SCIENCE AND INFORMATION THEORY LEON BRILLOUIN Adjunct Professor of Physics, Columbia Universit~", 4\1.ember of the National Academy of Sciences
ACADEMIC PRESS INC• PUBLISHERS • NEW YORK 1956
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319 319 320 322 324 328
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337 337 338 339 342 343
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P=2n, ll H3 (
1. J)
nonyqaeM HHcpOpMauHIO
I= K1n P= Kn Jn 2. Mhl iKenaeM OTOiKJJ.ecrBHTh H 6epeM:
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(qHCJIO J.IBOHtJHhlX e.llBHHU)
( 1.3a) TaKHM
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B llHOHQHblX
I= log'l P.
e.llHHHuax
paena
(l.3b)
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CHCTBMht B,llHHHU
21
B
Kattecrse npuMepa paccMOTpHM KOno.u.y HS 32 paanHllH&Jx Ka pr, 113 KOTOpbIX JJ.OJiiKHa OhITh Bh16paua o.u.ua. Ha ocuosaHHH ( 1.3b) Mhl CKaiKeM, ttTo I= 5 .n.souttuhlx e.u.uuuu, TaK KaK 32 2is. Ho B03hMeM reneph 11ee KOJIOJlhJ, no 32 KapThI B KaJK110A. EcnH Mhl Bh16epeM .u.ae KapThI, no 011uoA 113 KaJKnoA l{OJIO.U.hl, TO iKeJiaTeJihHO, tfT06hl Mhl Mor JIH CK338Tb, •110 HHQJOP· MaUIHI npH 9TOM B,U.BOe 60JihIIJe. 06mee '-IHCJIO B03MOiKHOCTe~ paeuo
=
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I= KlnP= Kin (P 1P 2)= KlnP 1 +K1nP2
a
H, raKHM o6pa30M, I= 1 JJ.BOH'-IHhtX e.llHHHU. YiTaK, norapMcpMMllecKoe onpe.u.eneuHe nucpopMaUHH npe11cTaensieTCS1 pasyMHhIM. Jlpyrast CHCTeMa e,u.HHHU MOiKeT 6h1Tb BBe,llena, eCJIH Mbl cpaBHHM unif:Jop.MafittlO c TepMo.u.11uaMHllecKott aHmponuea 11 6y.u.eM l13MepSITh o6e BeJIHlll1Hhl B O.U.HHX H Tex JKe e,U.MHHU3X. 3uTpOTIMSI (CM. rnasy 9) HMeeT pa3MepHOCTb 9ueprMH, )leJieHHOA Ha TeMnepa Typy. J].JISI 9HTp0nHH MMeeTCH cpopMy Jia DOJlhll,MaHa, oqeuh cxo.u.uasi c ( 1.1) u co.u.epiKautasi K09cpcpttuueuT
k
=
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3Ta
(1.4)
KOHCT8HTa H3BeCTHa KaK DOCTOSIHHaSI BOJihUMaua (CM. (9.15) ). Ecnri Mhl aae.u.eM 9TY nocTosiuuy10 k BMecro K B (1.1 ), TO Mhl 6yJleM H3Meps:tTh HHQJOpMBUHIO 8 e,U.HHHuax 3HTponMH. Mhl MoiKeM c.n.enaTh em.e OJJ.HH war H Bhlopa Th eJJ.MHHUhl TaK, llT06hl KaK 9HTpOrIHSI, TaK H MHcpopMaUHSI 6hIJIH 6eapa3Mep· HhlMH BeJIMllHHaMH H npencTaBJISIJIMCh OTBJielleHHhIMH llHCJJaM11 1) • .IlnSI aToro nyJKuo usMeps:tTb TeMnepaTypy B e1111u11uax sueprHH. Oohllluas:t cTorpa11ycuaS1 wKana npHMeHHMa, ecnM k HMeeT llHC· neuuoe aHalleutte (1.4) H paccMaTpHaaeTCH KaK oTaneqeuuoe tlHCJIO. IlpH 9TOM OTHOIIJeHHe e11HHMU uaIIJHX llBYX CMCTeM T3KiKe 1
)
D. E. B e I I, J. Appl. Phys. 23, 372 ( 1952).
22
onPE~EnEHHE
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(1.4a)
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=
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(1.5)
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I= Kln Na= KO In N.
Mhl Mo>KeM Bhlpa3HTb ( 1.6) a .LJ.BOHlIHblX I=
(1.6)
e~wuwuax:
a log2 N,
(1. 7)
TaK KaK
(log2 N) (In 2) =Jn N TaK, eCJIH
1.V = 2, Mbl I= a
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"
1 K= In 2 ·
HMeTb 20 pa3JIHlJHbIX HCXO.UOB li
~BOHQHbIX e.LJ.BHHU HHcpOpMaUHB.
( 1.8)
JierKO noKa3aTh, QTO Mbl nonyQHM TO >Ke KOJIHQeCTBO HH· cpopMaui-u1, ecJIB H3MeHHM ocuoeaHHe N H cooTBeTCTBeHHO lIHCJIO uwcpp. IlycTh, uanpHMep, BhIOpauo ocuoaaune N' = 8. Tor .LJ.a .UJJSJ aanHCH paccMa1p0eaeMoro QHCJia noTpeoyeTCSJ ecero O' = 0/3 uwcpp, H
=
I =0' log2 N' qTO cornacyercH c (1.8).
=
~
· 3 =0,
( 1.9)
41
HH~OPMAUHH
H
PaccMorpHM .uecH'fHqHoe qHcno HS
N= 10
23
AJI~ABHT
0 10
uHcpp:
/=010 log9 10=0 1o3,32
H
3,32 .ua. e.u. B 3,32 pasa
(l.10)
,n:B. e.u:.
OJI.Ha .u:ecHTHqHaH uwcppa .uaeT
HHcpopMauHw; .u:eoHq ..
HaH
60JibWe
38ITHCb
llHCJia
Tpe6yeT
UHcpp,
lleM
4. Hnct>opManHH H anct>aeHT DOJiblllaB qaCTb HCOOJib3YeMott HaMu HHcpOpMauwu coo6maeTCSI rrocpe.ucTBOM H3b1Ka. HBJIHIOTCH B
B
OCHOBHhie
seyKH
(Ha3hlBaeMble
peqH
CJIOBa
COCTaBJieHbl
OHChMeHHOlt
ITHChMeHHOe
npe.UJIOiKeHHe
UMH, co.u:epiKarn.ettCH B .uaqa
ycTHOtt pellH 9JieMeHTapHbIMH CHMeonaMH llaCTO li3
H TlO.UCllHTaeM KOJIHqeCTBO HHcpOpMa-
9TOM
8Ta
rrpe.U:JIOiKeHMH.
HMeeT
YBH.U:MM,
HeHSBeCTHO 0
OOJIHOe H CTporoe M3-3a
a
6yKB. PaCCMOTpHM
6011bUIOe .npaKTHlleCKOe 3Haqeuwe; .u:po6Ho oocy>KJleHa 1\. llieHHOHOM 1) H MHOrHMH Mbl
cpoHeMaMH),
pemeHHe
OTCYTCTBHH ITOJIHblX
CJIO>KHa51
OHa
OhIJia
.upyrHMH.
3al(aqu
JlO
sarroKaK
CHX
nop
craTHCTHqeCKHX .u:aHHblX
SlShlKe.
MbI
MO>KeM
HeOOXO.U:HMO ancpaBHT
paccMaTpHBaTb
t5yKBbl
KaK
CHMBOJibl, KOTOpble
BblOHpaTb .U:JIH ITOCTpOeHHH npe.uJIO>KeHHSI.
co.u:epiKHT
27
CHMBOJIOB:
MeiK.uy cnoeaMH. EcnH 9TH
27
26
IlOJIHblit
6yKB ITJllOC npoMeiKyTOK
CHMBOJIOB paeHOBepoHTHbI anpMop11,
TO Mbl MO)KeM CKa3aTh, qTO HHcpopMau11H, co.u:ep>Kaw.aHCH B npe.uJIO>KeHHH HS
Q tJyKB, COCTaBJIHeT:
I= a logg 27
i= 9TO
logg
27 == 4, 76
COOTBeTCTByeT
0.u:HaKO
nonyqeHHOe
rrpHMOMy TaKHM
(1.11)
.UB. e.u:.
.U:B. e.LJ.. Ha oyKBy. npHMeHeHHIO
rpopMyJihl
(
1. 7).
o6pa301\f pemeHHe Hey .U:OBJieTBOpH-
TeJibHO, TaK KaK pasJIHqHhie 6yKBbl BCTpeqa!OTCH B HSbIKe c He· OJlHHaKOBbIMM anpwopHbIMH BepOHTHOCT51MH ( TaO.lJHUa IlycT&
. . . , 27), 1
p1 KaK
arrpwopHaH BepostTHOCTb yKasaHo
e
Taonwue
1.1.
j-tt
t5yKBhI
Cpe,n:HHH
1.1 ). (j 1, 2, ...
=
MHcpopMaUH51
C. E. Sh an non, \V. \Ve aver, The l\lathematicaJ Theory of Communication, U. of IHinois Press, Urbana, 111. , H) 19. )
24
frn.
ODPB.llBnBHRB HH+OPMAUMM
Tat5nMua
l
1.1
BepoHTHOCTL noueJ1eeus p H seaqeeHR-log 10 p JI.JIB 6yKB aer JIHACKOfO RSblKa
CHMBOJI
BepO.HTHOCTb
p
-
logto
Ii
P
CHMBOJI
I I
IlpoMemyroK E
T
0 A
N I
R
s
H D
0,2 0,105 0,072 0,0654 0,063 0,059 0,055 0,054 0,052 0,047 0,035
0,699 0,979 1,143 1,184 1,2 1,23 1,26 1,27 1,28 I 33 ' I,46
BeposrTHOCTb
l
I
c
FU M
p YW
a B
v
K
x
JQZ
I I
p
- log10 p
0,029 0023 ' 0,0225 0,021 0,0175 0,012 0,011 0,0105 0,008 0,003 0,002 0,001
1,54 1,64 lt65 1,68 1,76 1,92 1,96 1,98 2,1 2,52 2,7 3,0
Ha O.llHY 6yKoy, cornaCHO llieHHOHy, aanHrneTCSI KaK 1=~1
i=-K _!.Pi lnp1.
( 1.12)
j=l
Oocy .llMM npMMeHMMOCTh 9TOJ1 cpopMynbl.
5.
He4JopMaU.HB, COllep)f(aU(aBCB B COBOKyDHOCTH CHMBOJIOB c pa3nHqHMMH anpHOpHYMR eepOHTHOCTHMH npe.llnOJIO)f{MM, QTO
M
Mbl npMMeHHeM
pa3JIMqHbIX CHMBOJIOB
1, 2, ... , j, . .. , M, KOTOpble MMeIOT anp11opHble Ph p2, ... , pj, ... , PM cooTBeTCTBeHHO. H11KaK11x
oepoHTHOCTH
.D.pyrMx ycno.. BH" 11n11 orpaHnqeHMlt OTHOCMTeJibHO npMMeHeHHH 9TMX CHMBOJIOB He HaKJia.llhlBaeTcst.
(1.12),
Mbt )f{enaeM BbieecTn meHHOHOBO paaeHCTBO
COr JiaCHO KOTOpOMY DOCJie.llOBaTeJILHOCTh M3
0
CHMBOnOB
co.uep>KMT HHcpOpMaUMIO i=M
l=Oi=-GK
2:p
1 lnpJ,
i=l
(1.13)
HllOPMAlUHl C PA3HhlMM AilPHOPHbIMM
BBPOHTHOCT~MH
r~e i--cpe.llHHH HHcpOpMaUHff Ha CMMBOJI M
i=M
!
Pj= 1.
j=I
HaqHeM c npocTott sa.n:allM, KOtopy10 sareM nerKo 0606IllMTh. PaccMOTpHM ancpaBHT H3 llBYX «6yKB» - roqKM H rMpe, KaK B TeJierpacpHM, HJIM 0 M 1. PaCCMOTPHM IlOSHIJ,Mtt (cells HqeeK), H nycrh N 0 noa1111.wit co.n:epiKaT 0, a N1 CO.llep>Kar 1, T8K qTO No+ N. TaKHM o6pa30M, BCe Il03HUMM aanoJJHeHhl. Tor .u.a BepoHTHOCTh nosMuHM, CO.llep>Kamelt 0, 6y .n:er
a
=a.
No
Po=a, H aepOSITHOCTL roro, QTO 003MUHSI CO.llepiKMT
( 1.14) 1,
ecTL
Ni
Pi=a,
( 1.15)
Po+P1=l,
(1.16)
npMQeM
T8K KaK BepOHTHOCTb Toro, lJTO B HeKOTOpoit fl03MUMM co.u.ep· iKHTCH JlHOO 0, JIMOO 1, paBHa e.n:MHMUe. Tenepb Hait.n:eM lJHCJIO cnoco60B sanOJIHeHMSI Ka:tK.n:ott M3 Il09MII.Mit JIM60 0, JlHOO 1 (HO HHKOHM o6pa30M He o6oMMH CHMBOJiaMH OllHOBpeMeHHO). 9ra san.aqa B TOQHOCTH Ta iKe, QTO B CTaTMCTHKe epMH. YHCJIO cnoco6oB sanOJIHeHMff Il03MU1Ifi paoHo qMcny cnocoooa sanonHeHHSI N 0 nosMuMtt HYJIHMH, TaK KaK, ecnw Mbl y}f\e pacnpe.n:enHJIM N 0 Hynefi, ro ocran&Hhle Ni nosHuwtt .u.on>KHbl co.n:ep:tKaTL no e.n:MHMue Ka>KJJ.aH. Ho QMCJIO cnoco6oB sanonHeHMff N 0 noaHUMit HYJIHMM pasHo llMCJiy nepecraHoaoK (c DOBTOpSIIOlll.HMHCSI 9JieMeHTaMH) H3 no No:
a
a
a
QI P= Nol Nil.
(1.17)
3TO eCTb QMCJIO C0061UeHMit H3 Q CMMBOJIOB .U.BYXOYKBeHHOrO ancpaswra, OII.MH M3 KorophtX BCrpeqaercH N 0 paa, a .llpyrott Ni pas. llnH OllHOro MS raKMX coo6meHHit cor nacHo ( 1.1) HHcpopMauMH paaHa
I =Kln P=K(ln O! - In N 0!-1n N 1!).
(L18)
26
OJIPE,ll,EJIBHHE IUl4>0PMAUlUI
a,
ECJIH coo6m.eliHe .nnHHHO H No H Ni .nocTaTOlfHO BenHKH, To norapwMhl cpaKTOpHaJIOB MoryT OhITb BhlpaiKeHhl np116JIHiKeHno Ha ocHosaHHH cpopMy nbt CTupnuHra:
In Q! ~ Q (ln Q- 1). 3ra cpopMyna HHe ,nJISl Q
Jl&eT,
> 100.
l ~ K [0 (In Q -
(1.19)
KaK nseeCTHO, oqeHb xopowee np1101n1)Kel1TaK, ecnH 1, No> 1 H Ni> 1, TO
a>
1) - N 0 (ln N 0 -
1) - Ni (In N 1 -
1) ] ,
(1.20) CHOBa npHMeHSH1 B BHJI.e
I
paBeHCTBO
va ( -No ,a
~--n'
a= No+ N., InNo -
a
nepenurneM ( 1.20)
+ -NiaJ nNia· -)
(L21)
IlO.U:CT3BRS1SI ( 1.14) H ( 1.J 5) H ,nenst Ha Q, nonyqaeM;
i=
~ ~ -K (Po 1npo+ p 1 1npi),
(1.22)
r .Lte i - HHopMaUHSl Ha CHMBOn coo6rueHHSl. 3To H eCTb cpo pMyJia llleHHOHa JI.JIH cJiy1.1aH Jl.Byx CHMBOJIOB. 3aMeTHM, qTo p 0 11 p 1 MeHhUie e.LI.HHHUhl, HX JIOr~pHcpMbI 01p11uaTeJihHhI, a noTOMY cpopMyna ( 1.22) .naeT rronoiKHTenhHOe sHaQeHHe .LtnH i. 06oomeHHe Ha cJiyqafi't 6oJiee lfeM .n.eyx CHMBOJIOB noJiy .. 4aeTCSI nerKo. 06osHaq11M qepes Ni, N 9, ••• , Ni, . .. , NM llHCJia CHMBOROB M pa3JJHqHbIX THfIOB H Bbl6epeM QHCJIO ll03HUHJI
(1.23)
Onpe,ll.eJIHM eepoHTHOCTh
j-ro
CHMBOJia
( 1.24)
5]
27
HHonee OJ.I.HOto CHMBOJia), cocTaBJISJeT:
QI
( 1.26)
P= }= - .M
II NJ' j=I
cI>opMyna (1.26) stBJISieTCH npHMblM ot5o()meHHeM (1.17). Mbl no· ny11aeM .n:nsi 11HcpopMau11H, co.u.epiKameltcsi e OJ.l.HOM HS cooc>meuutt: J=M
l=KlnP=K[ln(O!)-
!
ln(Ni!)]~
j -I
j=M
~·K(OJnO.
!
NilnN1).
(1.27)
j~I
PaBeHCTBa B 9TOM BbipaiKeHHH COOTBeTCTBYIOT (1.18), ( 1.19) H ( 1.20), noJiaraSJ no ... npeiKHeMy, qTQ H Ni .lJ.OCTaTO'-IHO Be# JIHKH, TaK qTo npnMeHHMa KHYIO ceo6oLty BhI60pa, BeLteT K yMeH&WeHHIO HH$OpMaUHH. PaccMOTpHM c11cteMy, npe.n.cTaBJISJIOlll.YIO P pas11HQH&1x B03· MO)KHOCTeti, KOr)la BCe nepeMeHHble CBOt.50.llHhl. Kor .ua Mhl HaJiarae:u Ha nepeMeHH&Ie orpaH11qeH1111, yMeH&ma10m0e ceooo.n.y BhI· OOpa, 3TH ycJJOBHH HCKJIIOqaIOT HeKOTOpbie HS paHee cymeCTBO• easlllHX B03MOmHocreti. Hoeoe q11c.r10 BOSMO>KHOCtett P' c orpan11qeHHHMH JlOJl)KHO 6h1Tb, oqeBH.UHO, MeHbWe, lleM nepBOHaqaJibHOe P, H D09TOMY Mbl .D.0Jl)l(Hbl IlOJiyqHTb HOBOe SHaqeHHe HHcpopMaUHH I' 6es orpaH11lleH11tt: P eosMomHocrelt, I= K Jn P; c orpaHHlfeHHHMH: P' B03MOlKHOCTett, npH p p H
< /:
i'=KlnP'
OTpHuareJibHOii UCHHOCTH HHcpopM8UHH, TO ee ecrecrseHHO 6blJIO 6bl npHDHChlB8Tb JIOlKHOA HHopMaUHH, Ha OCHOB8HHH KOTopoa tteJ10seK (HJIH 88TOM8T) MolKeT npe.nnpHHffTb JleACTBHff, Heuenecoo6pa3Hble npH J{aHHbIX .n;eAcTBHTeJibHhlX o6CTOHTeJibCTBax. 3TOH roqKH apeHH.R IIOH.flTMe o ueHuocrH HHct>opMauHH He TaK ylKe HeoC.RJaeMo. UeHHOCTb HHcpopMaUHH MOlKHO onpeJI.eJIHTb qepea npupa14eHUe 8eponmHOCmU OOCmU:JJCeHUR qeAU, .nns KoropoA co6HpaeTCR BHopMauus .. TaK, .IJ;JI.fl qenoseKa, npecneJI.yiomero npecTynHHKa H HMe1omero ue11bIO noAMaTb ero, HHopMaU.HR, flOJIJlleHH8.fl OT npoxo>KHX, 0 HanpasJieHHH, B KOTOpOM CKpb11IC.R npecryn.. UHK, 6yJI.eT HMeTb IIOJIOlKHTe.'IbHYIO ueHHOCTb (KoTopyIO MOlKHO Bblpa3HTb 'JHCJioM), ec.rzH Hanpas,'IeHHe yKa3aHo npasw~1hHo, H orp11uareJibHy10 ueHHOCTb, ec.'IH yKa3aHo nolKuoe (1-1anpuMep, nporHsononolKHoe) uaupasneHnc.
c
(llpu.M. nepee.)
rJIABA2
TIPHMEHEHHE OTIPE}lEJIEHHR H 06lllEE 06CY>KJlEHHE
1. OnpeAeJieHHH B
nepsoA r nase
6L1no
noKasaHo,
t1To
Mepa
11HcpopMan1111
KaK CTeneHH HeonpeJJ.eJieHHOCTH, cym.eCTBOBaBweA JJ.0 Toro, KaK BblOOp OblJI Cll.eJiaH, SIBJISieTCSI
TOllHOtt,
HO
orpaH11lleHHott. TaK, Hanp11Mep, qeHHocmb
na
no
HeOOXOll.HMOCTH
HHcpopMan1111 He Mor-
ObITb BKJIKlt:feHa n 9TY Mepy. DblJIO TaK>Ke TICK333HO, llTO eCJIH B03MOiKHhle BbIOOpbl HMeJIH
uepaBHble anp11opHble nepOHTHOCTH, TO 9TH
BepOSITHOCTH MoryT
paccMaTpHBaTbCSI KaK orpaHHlleHHH Harnero
BhlOOpa,
llTO BeJI,eT
B KOHeqHoM Cl.leTe K yMeHbllleHHIO KOJIHqeCTBa HHcpopMaUHH. TaK, ecn11 anp11opHb1e nepoHTHOCTH JI,JIH CHMBOJIOB ( 1),
. . . , U), • . .
paenhl
cooTneTCTBeHHo
p., p 2,
..... ,
(2), ...
p1, ..... ,
To
KOJIHlleCTBO HHcpopMaUHH Ha CHMBOJI, KaK OblJIO noKasaHO panee ( Cl\I. ( 1. 12)) , pa BH0
i=-K
2>j 1i1p1.
(2 . 1)
i 9To Bb1pameH11e OhlJIO nonyqeHo 113 cpopMy.11&1
JJ.JIH HHcflopMaUHH
Ha CHMBOJI B cnyqae, Kor .ua BhIOOp He orpaHHl.JeH: (1.~)
JJ.JIH
m
paBJIHqlfbIX paBHOBepOSITllblX
CHMBOJIOli, r ll.e Q -
l.IHCRO HCOOJlhBOBaHH~X CHMBOJIOB {M~ rnaBhl
1
Ha
m).
l1CXOJJ.SI H3
0JJ.HaKO l\tHOrHe aBTOphl, HCXOJJ.HOe
COOTHOWeHHe
nocpeJJ.CTBeHHO.
B
(2.2),
saMeHHJIH
Mbl BhlBeJIH
oto6eHHOCTH
(2.1 ),
A1
H3 KOTOporo
paaJI,ena
cpopMyny
WeHHOH,
(2.2)
OOIItee
5
(2.1).
opaJIH
sa
c ..ieJJ.yeT He·
2]
CBOiiCTBO
A
33
m
llef:4CTBMTeJibHO, TIOJIQ}l{I1M, lfTO Mbl MMeeM pa3JIHlfHhlX CHM• BOJIOB c OAMHaKon~MM aupuopHMMM BepoHTHOCTHMM. Tor4a
P1::..::. P2 = 11 H3
(2.1)
· · ·=
I
(2.3)
P1· = · · ·=Pm= -m ,
noJiy1.1aeM:
I I -ln -
i=-K ...... ~ m
m
I
=-Kln--=Klntn m m '
(2.4)
q,'leBOH
a 3TO M eCTh paBeHCTBO (2.2). PaccMOTpHM Tenepb HeKOTOpbIC CBOtiCTBa COOTHOllleHHSI
2.
CeoiicTeo
(2.1 ).
A
EcJIH Bh16op pa36HT Ha 4Ba noc.11e,U,oeaTenhHhIX BhIOopa, TO nepBOHa4aJibHaSJ 111-HpOpMaUMH oy ,UeT BblpamaTbCH BSBemeHHOf:4 cyMMof:4 ,U,syx 1.1acTH1.1Hh1x HHcpopMauwf:4. PaccMOTpHM np11Mep. BepoHTHOCTh nepexo,ll,a us T04KH 0 e T04KY A paBHa 1/2, ws 0 B R - 1/3 H 11s 0 B C - 1/6, KaK noKa3aHo cxeMaTw 1iecKH na pwc. 2.1. PaccMoTpHM, OAHaKo, ,L1,11arpaMMY pwc. 2.2. Ha 9TOM
0
8 P0c. 2.1. BepoHTHO· crn
nepexoJJ.a
TOQKH
0
B TOlJKM
B
H
c.
n3
A,
P11c. 2.2. BeposnHocrM nepe· XOJJ.a M3 TQljKH 0 B TQljKff A, B H C c npoMelKyro11ttoA TO'lKoti M MeiKJl.Y TOllKo.H 0 If T04&.aMH
B
M
c.
pwcyHKe B&16paHa npoMemyT04Ha5l TOl.fKa M TaKHM o6pa3oM, 1.1To BepOHTHOCTh nepexo)la ua 0 B M TaK>Ke paBHa 1/2. 3To 3Ha4HT, 1.1To aepoHTHOCTh nepexo4a 11s M e B 40JiiKHa OhlTb paBHa 2/3, TaK KaK nepBOHa1.1aJibHa5l Bepo~THOCTb nepexo)la IIS 0 s B paBHa I /3. AHa.11or114HO, seposiTHOCTh nepexo4a H3 M B C paena 1/3.
2 J!.
BPD "IJIIO';)H
34
2.1
0603Hal.IHM MHcpOpMaUHIO, COOTBeTCTBYIOIUYIO pHC.
I (~ , ecTh
1
[ r J1.
HPHl\IEHEHHE OnPEJI.EJJEHHH H OBIUEB OBCYiKJI.EHHB
~,
!).
llJJS1
I ( 21 , 21 ) , I
2 \,:r, I
6blrh
I) . 3
a
DoJIHafl
O)l.u11aK011a:
pnc. 2.2
1rntfiopMaUHSI )l.JJSI
HHcpopMaUHSI
JI.JJ5l
HHcpopMaUHSI
I (-} ,
JI.JJH p11c.
! ,-Ir). 1,13
1.1epe3
nyrett OA, OM
MB, MC
nyTefi
2.1
2
H
2.2
ecTb
JI.OJJ:tKHa
p11c. 2.1 11MeeM:
1 ) - - K (-~ In J_ __!_ I ( J_ 2'3'62 2 HJIH
1u.!.!)= =-
K [ ; In
P acKpbma5l
~
+ -;- · ~ In
n·~-) + ~ u·~ )]. 31_ In
(2, 6)
H rpynnttpyst, nonyqaeM:
_!_ 1 ) ii ( _I 2' 3' 6 -
+ 1 In 21- ) + -21 ( 32 In 32 + 31 ln 3I \1) = = - K (-} In ~ + ~ In -} ) + -} [- K ( ~ In ~ + } In ; ) ] = =I ( ~ , ~)+-}I (~ , -} ). (2.7) 1 1 = - K [( -~f In -~;r
IlOCJie,ll.Hee Bblpa:tKeHHe COOTBeTCTByeT pHC.
2.2,
,ll.Jlfl
poro nOJJHafl HHcpopMaUHfl JI.OJJ:tKHa paBH51ThCfl B3BerueHHOfi Me 11ucpopMau11n JI.Jlfl nyTeii
OA,
OM
11
MB, MC.
IUHti pesy Jlh TaT, flBJJfllOW.HfiCfl C.lleJI.CTBHeM Tor 0 cpopMaUHfl -
cpaKTa,
cyM-
06·
llTO HH-
JJOrapHcpMHqecKafl cpyHKUHfl.
3. CeoHcTBO
3To -
KOTO-
B
CBOHCTBO
B
Bb1pai*aeTcst HepaBeHCTBOM, KOTophlM HaM npe.n.-
CTOHT nOJih30BaThCS1 B JI.aJihHettrueM.
YTo6hl
paneHcTBo, paCCMOTpHM cpyHKUHIO
lJf
= _!p1lnq1, j
no.i1yqHTh
9TO He-
CBOU.CTBO B
3]
35
r.ue Pi - eepoffTHOCTh j-ro Bh16opa, a qi MoryT paccMaTpHeaThCH KaK ixpyrasi COBOKYIIHOCTh nepoSITHOCTeti, TaK QTO
(2.8) 3aniinneM
qi=Pi+u1=Pi(1
+~)·
Tor.n.a uj yixoenernopmoT ycnoBHIO
.!iui = 0.
(2.9)
j
Cpanmrnan rpaHKH In (I
Pttc. 2.3. fpaqmKM u/p
+;) H u/p
fl
ln (1
(pHC. 2.3),
+ u/p)
Mbl
BH.D.HM,
B lj>yHKU.HH OT u/p.
lfTO
u....- ;?: Jn ~I 1
p
11es!le, Q•
r.o.e cymecTeyeT
'
+ u) -P.
In\ 1 + ~).
(2.11)
IlpHMenn11 (2.11) K (2.10),
36
TIPHMBHBHHB onPRARJJRHHf:t "
OSlllBE OSCYiK;J.HHHB
[ r JI.
2
nonyqaeM:
~ p·lnp·+ ~p.1n(1+1!_j)~ J J .... J p· ~
.'. J
. J
I
~ '1 p . In P1
~. '
-1-11
(2. 9), HaXO.LJ.HM:
Ilo9TOMy) HCil0Jlb3YH
i Pi
Inq1 ~ .!/i lnp1 .
J
(2. 13)
(2.12)
J
J
YMHOiKaH CBOitCTBO
~ p . "!i "-' . 'P. I
j
K,
ua -
(2.13)
nonyttaeM uepaeeHCTBo,
Bhlpa>Karomee
-K 2>11nqi~-K !Pi lnp1.
(2.14)
B: I
I
4. · l1HcpopMau.11s:r CTeH paBHhl:
I
HMeeT
CeoucTeo MaKc11MyM,
- - · · ·-1, - 2 P1-p
C Kor .na ece
1
-p m---. i-···m
m
BepostTHO·
(2.15)
Llnsi llOKaaaTeJibCTBa saMeTHM, npeiKlle ecero, llTO HHcpopManns:r s:JBJIHeTCH cpyHKUHeH TOJibKO (m - 1) He3aBHCHMhlX nepeMeHHhlX, TaK KaK }==m
j=m-1
!Pi= 1,
HJIH
Pm= 1 -
j-I
BoshMeM
e
!
P~ ·
(2.16)
j=I
Kat.1ecTee
(m -
1)
HeaaeHCHMhlX
nepeMeHHhlX
p.,
p 2,
••• , Pm-t· l1ucpopMan11H I (p 1, p 2 , ••• , Pm-t) 11MeeT MaI B npocTeth11Hx rpacp11qeCKH. B cJiyqae .a.eyx B03MO>KHocreH, Mbl HMeeM:
I= - K (p1 ln P1
[ rJJ. 2
06ll{EE OBCYiK.ll.HHHH
c;1y1.1asix noKaaaHo npn p 1 P2 === I,
-f-
+ P2 ln P2).
(2.18)
Ha pHC. 2.4 I nocrpoeHa KaK cpyHKitHH p 1 (HJIH p2). I JI.OJI)l{HO paBHHThCH Hy1uo npw p 1=0 H p 2 =1. B c11.11y CH:\1MerpHH
I JI.OJIJKHa HMeTb .MaKCHMYM npH p 1
=
~
=
p~.
3TO
H
MaKCHM3JibHOe 3HaqeHHe paBHO
I!JISI rpex B03MOiKHOCTeH
Pi -1- P2
+ Pa =
H
I = - K(p1 lnp1
+P2
(2.19)
I,
ln P2 +Pa
lnp3)
HOCTpOHM JI.ByMepHyiO )I,HarpaMMY (pHC. 2.5) . .llByx U3MepeHHti JI.ocraro11Ho, raK KaK TOJihKO JI.Be 113 rpex eepostTHocrett p 11 p 2 ,
I Kln2 f '2
C2 •
Pi
Pz
0 P2
Pwc. 2.4. VfaKJJ.EHHE
H'.alKJlOI:\
H3
JlByx
nepeMeHHblX.
TaK,
Mbl
MO>KeM
fr JI. 2
nMeTh JlJHI
= i, y = j. Onpe,LI.eJI0M p (i, j) KaK nepoHTHOCTb x = i 11 y = j e OJJ.HH H TOT iKe MoMeHT. Tor.LI.a
f~ x
. Mo,renra
*foro 1 1.1To
~>(i, J)= 1.
(2.21)
i, j
Pi
EcJIH
ecTh nepOHTHOC'fh Toro, QTO
Ol 3Ha tteIU:UJ
J'),
x= i
(aHe 3aBHCHMOCTH
TO j=n
P; =
.~> (l,
(2.22)
;).
j=l nepOHTHOCfh
1\HaJIOfHqHOJ
Toro,
4TO
y =j
(He paCcMaTpHBaH
x), ec1b i=m
(2.23)
P1= !p(i, j). i=l
MbI
HMeeM CJleJI.yrou1ue cooTHoweHusi:
,,
m
n
~
1-Pi = l
~P;= l, i=I
ITepet:t.n.eM ruetl
H
pa3JIHqHhIM
(2.24)
i, j
j=I
K
!PiPj=l.
Bhipa>KeHHHM JI.JIH nHcpopMau0n, cooraeTCTBYIO· eepoHTHOCTHM.
CoaMecrHaH
HHcpopMau11H ecrh
J(x, y)=-K,!p(l, j)lnp(i, j).
(2.25)
i. i
HHq>op~1aUJH1 .LI.JIH
x
B OT,LI.eJihHOCTH
i=m
l(x)=-K !P;lnp1 =-K !p(i, J)lnp1, i=l
e JI.JHI
y
(2.26)
i,j
B OT)J.eJibIIOCTH
n
1(Y)= - I< !Pi lnp1 = - K !p(i, ;) lnp1 • j= 1
i, j
(2.27)
YCJIOBHA~
CpaaHH!\t cyMMY
41
HH~OPMAUHH
I (x) +I (y)
c
I (xJ y).
J13
(2.26)
(2.27)
H
(2.28)
l(x)+I(y)=-K_!p(i, J)Jn(PiP1 ). i. J
B,
JlerKo noKaaaTh, HCIIOJih3YH caof.lcrBo
l(x,
B
y)~l(x)+I(y).
(2.14) q1
caMoM 11.ene, saMeHHst B
J (x, y)== -K ~p(i, j) lnp (i, j) i,
qro
Ha
(2.29)
Pl p1,
nony 1iaeM:
~
J
:s;;- K _!p (i, j) ln (Pi p1) =I (.x) +I {y). i. j s:ICHO, tITO Ha.11araeT
orpaHH4eHHe,
BhI3hlBaeT TOJihFKe JipyrHe eepo51THOCTM. Be11nqnHa
! p (l, J)
paBHa B€po51THOCTlt TOro, tJTO
x= l
JV151
mo6'oro
j
J'· C pj =
I (x).
9TOti BepostTHOCThIO CBH3aHa HHcpop'.'4aUHH
,! p (i, j) -
sepo51THOCTh Toro, tJTO y
=j
Be11HqnHa
mo6oro
.D:JISI
x.
i
Csfl3aHHast c 9Tott eepoflTHOCTbio 11Hcpop~au1151 eCTb
B
I (y).
npeJlhlJlymett r 11aBe 6h1110 HafiJI.eHo, qTo
I (x) +I (y) ~I (.x, y). 3TO HepaseHCTBO CJie}lyeT
H3 Toro cpaKTa, qTo npH HaJIO)KeHHH
Ha cHcTeMy orpaHnqeHntt MfnpopMaun51 y6b1BaeT. Jl11g
H.ocmb eecTHO,
no.ny4eHM51
P; (}), qro
nHcpopMaUH51
T.
e.
x = i.
I .x (y).
paseHCTBa
Mbl
Bepo51THOCTh
c
9TOil
B3HJIM ycAOBH.yJO BepoHmToro, qTo y
eep051THOCTblO
= j,
KOrJla ns-
CBH3aHa yCJIOPH3Sl
44
H3BMTOqHOCTb B 513hlKB
[rn. 3
CBH3h MemJly sepoHTHOCT5IMH Bb1pa>KaeTc5I cooTHollleHHeM
P;Pi U)=p(l, j), 11, onpe.uen11s yc.noBHYIO 11HcpopMa1uno, KaK
Ix (y) =-K
_!ip (l, J) lnp; (J), i,
Mbl IlOJI)'llHJIH CJie.nyromee
J
paBeHCTBO~
2. KoppenHUHH n
H3bIKe
}{oppeJUIUHH HMeeTC51 BO ncex 513bIKax 1), . raK KaK, ecnH I10· StBJI5IeTCH HeKOTOpa51 OyKBa, TO sep05ITHOC'fb I10HB.netH151 3a nett HeKoTopotl .upyrott 6yKBhI He pasHa ee anpHOpHotl seposrr1-1ocrH. TaK, B aHr JIHtlCKOM Sl3bIKe, ec..iu1 ,llatta oyKBa t, TO BepoRTHOC'fb TOfO, qTO Henocpe,llCTBeHHO flOCJie,llyIOU~etl oyKBOti oy )let h, ropa3)l0 OOJibWe sepoHTHOCTH, qro c11e.uy10ruetl oyKBOtl oy .uer n. AHanorwtfHO, ecnH JI,aHo cot1eraHHe tio, ro sepoHTHOCTh nons11eH11st n B Kaqecrse CJieJly~ett oyKBbl oqeHb BeJIHKa. Kop· pe11HUH5I no.uoo1-1oro po,lla onpe,llen5IeTC51 WeHHOHOM 2), 3) KaK u36blmottH.ocmb (redundancy). Ilp11MeHHM -renepb COOTHOllleHHSI npe,llhlJlYlllero pa3,lleJia K HCCJie,llOBaHHIO 11HcpopMaUHOHHOro co.uepmaHH51 c.0061ueHH5I. 3TH COOTHOWeHHH MOfYT I1pHMeHHTbCSI, ecJIH KoppeJIHll,HH HMeeTCR 1 )
B .nononHeHtte K npHBe.neHHblM HH>Ke nttreparypHhIM CCblJIKaM yKaJKeM CJiellyromHe KHHrH: F 1 et c h e r P r a t t, Secret and Urgent, Blue Ribbon Books, NY, 1939, 1942 (co.nep>KHT sepoHrHocrHblC .naHHble 06 aHrnHfiCKOM H3blKe); G. Dewey, Relative Frequency of Enqlish Speech Sounds, Harvard U. P., Cambridge Mass., 1923; 0. R. Z i pf, Human Behavior and the Principle of Least Effort, Addison - Wesley, Cambridge, Mass., 1949 [e 9Tofi KHttre H .npyrttx ny6nttKau,HJIX Uwncp llaer MHOfO KpHBblX, epo.ne npe.nct asneHHOtl Ha pHC. ( 3.1 ), )lJIH pa3~'IHq HblX Sl3bIKOB]. 2 W. We aver, The Mathematical Theory ) C. E. Sh an non, of Communication ( crp. 3-28) U. of Illinois Press, Urbana, 1949. llieHHOHOBa C9HTponHH Hm'.popMaUJflf> orom.necTBJIHeTCH c Hamett cMepoA HHqmpMau,HH», Ho He c
U)
ecTh neposrTHOCTh Toro, tfTO
j
nocJie.rr.yeT aa .rr.aHHhIM coqe'faHHeM b1 (N - 1). 3ro onpe.rr.eneHHe CXOJJ,HO c npHMeHeHHblM B npeJJ,bIJJ,YllJ.eM pa3,Aene, M Mbl HMeeM attanor11quyIO CB5I3b Me)lcJJ,y eepo5ITHOCT5IMM:
Onpe~eJIHM
TenepJ;>
cpeihtJOJO
u1u:jJop.Maqu10
Ha
cU.MBOA
46
[rn.
H3:SblTOlfHOCTb B ff3blKB
a
B nocJie)l.OBareJibHOCTH, B KOTOpott YLIHTbIBaeTCSI KoppeJISUlHSl Ha paccToHHHH .llO N CHMeonos:
FN= -K !P (b; (N -
1), j)
lnpbtKAY CJioBaMH) COJ.LepiKHT 5,5 6yKBhl, TO nonyqaeM npeJ.I;eJibHoe 3Ha4eHHe Fword 2, 14 J.I;B. en../6yKny. 9Ta eenHqHHa He ra >Ke, ttTO Fnm, TaK KaK HMeIOTCSI e1ue 11.ono.riHH· TeJibHhie orp aHHqeHH51, o6yCJIOBJieHHhie CB513HMH MeJK.ay nocne· AOBaTenbHblMH cnoeaMH.
=
4. HeKorophle 9KcnepnMeHThl J11151 no11yqeH11SI 6onee no11.po6Hb1X J.I;aHHhIX rrpeJ.I;JIO)l{eHa Hrpa: ()epeTC5I npeJ.I;JIO>KeHHe H yqaCTHHKy, He 3H8IOIIJ.eMy npe.anapH .. TeJibHO npeJ.I;JIOll HJIH «Her» npeJ.I;crasJI5ter KaK pa3 OAHY ABOH4HYIO eJ.I;HHHUY uttKJlbIH paa 3aHono H aaBMCMr, pa3yMeercH, or ncero upenmecrnytoIU.ero reKcra. TaKaH npouenypa upencras.1.Her, KaK HeTpy ntto BMnerb, nepeHoC B SKCrrepHMCHT MeronHKH KOJlHpoBaHHfl no llieHHOHy-auo, 1)
(llpu.At . .. nepeB.)
50
B3B~ToqnoCTh
B H3MKE
frJI.
3
flO-BHJ.I.HMOMy, B03M0jf(f10 COKpameHBe pasa R Tp11, KOTOpoe Mor110 ObI cne11aTb R3hIK 0011ee ToqHbIM. HynOpMaUH~ B ABOH4HblX eJI.H· HHuax, K KOTOpblM nepeXOJlHM, fiOJIO)KHB
K=log~e
( 4.4)
HJIH
C= lim log2 N(T) T-oo 1
T
'
(4.5)
) C. E. Shan non, W. Weaver, The Math.ematicaJ Theory of Communication, p. 7, U. of Illinois Press, Urbana, 111., 1949.
54 w
[ r JI. 4
nPUHUI1nbI KOJlHPOBAHHH
nponycKHaH cnoco611ocT& 6y.u.eT Bhipa'iKetta a JlBOH4HhIX e.llHHH·
T
Ilax e ceKyH.u.y, ecn11
nocne.u.y1011~11x
B
H3Mepetto B ceKyH.u.ax.
pa3.u.eJiax
9TO:tt r JiaBhI
Mhl 6y.u.e.M
o6cy)l(-
.uaT& npHfOJJ.HOCTb 9TOrO onpe}leJieHH51 H ycJIOBHe cywecTBOBaHH51
C.
npeJleJia
3aTeM
Mhl
noKan 1,
(4.13)
0Tp11uareJibHhlX HJIH KOMnJieKCHbIX KOpHeA:
X 2, X 3 ,
4.
••• ,
X0 •
OOcy>Knenne
TipeJlhIJlymast nocraHOBKa aaJla4H Jlatta nnepeble
We11HOHOM,
no.llpo6Hoe 06cyiK.lleH11e MO)f(HO Hat1r11 B JlHCcep1au1111
MaHJleJih·
1
opoTa ). Mhl HMeeM ooruee pellleHHe ( 4.10),
CO]lep)({aruee
a
UOCTOHH·
HhlX KO:Hpcp11uweHTOB Ak. 9r11 K09cpcp11u11eHTb1 onpeJlenmorc51 113 ttaqanbHhlX ycnoewit, Korophle Mhl ceAqac o6cyJlttM. PaccMorp11M
( 4.8)
npH MaJibIX 3Ha4eHl151X
fi=t 11 Mbl JlOJJn(Hht
nono)f(HTb
t.
0,D.l1H 113
HJJH
N (0)
'-IJieHOB COOTBeTCTByer
t-tj=O,
= 1,
TaK K8K
olleBH.llHO,
Mbl MO)f(eM npeJlCTaBl1Tb HeKoTOpble H3 cooomeHH~ TeJibHOCThlO
t = tj
llTO
o6me~ JlJIH·
np11 IlOMOlllH aj CHMBOJIOB ,lJ.JIHTeJibHOCTblO
tj.
tm
B6nhllll1e SHalleHHSI tf+tt ti+ 2 , •• ., He Moryr npHMeHHTbCst. 9ro yrsep)l{Jlett11e cnpasenn11eo JlJIH Bcex aua4eHH~ j. 8ro ycnoa11e y .D.OBJieTBOpSieTCSI, eCJil1 B3SITb
N(O)= 1, N(- l)=N(-2)= ... =N(-a+ 1)=0.
Mhl
noJiyqaeM, taKHM oopa30M,
JIHIOilll1'X
(4.14)
a Ha4aJihHblX ycJIOBHit, onpe}le ..
o npOH3BOJlhHhlX Ko9¢cp11uHeHTOB Ak. 3ttat1eH11st cpyHKUl1H
t
.llJISI = - a, - a - 1, ... HHKOf.a.a He BCrpeTSITC51 B ( 4.8) H He nrpaioT ponn n Halllef.t 3aJI.at.1e.
N (t)
1
B. l\1 and e J bro t, de communication, Ph. D. statis1ique de 1'Univ. de taKme T1ans. IRE (PGIT) )
Contribution a Ja theorie mathematique des jeux thesis, Paris, Dec. 16, J952, Pub!. de l'lnst. de Paris, vol. 2, N2 1, 2, pp. 80-102, 1953; CM. 3, 124 (1954).
58
(rJI.
nPHHunnhl KOAHPOBAHHS
4
k=a
N (t) === AiXf
+!
AkX~.
( 4.15)
k=2
llepBf,It\ qJien paCTeT 9KCflOHeHµ.HaJlbHO c t " ffBJHieTCH npeo6 .. Jia.u.aIOlll.H.M. OcraJibHbu: qJieHbl HMe101 KoneoateJJbHbifi xapa1KeM BO MHOfHX aanaqax HfHOPHPOBaTb KOJie6aHHSl H y ,ll.ep)({HBaTb B ( 4.15) TOJlbKO npeo6JJa)lalO~Htl t.tJieH:
( 4.18) r.n:e ~
= ln X
1•
VITaK, N (t) npeJlCTaBJisteT qHcJio coo61IIeuHtt, noJJuaH .ll.JIHHa KOTOpbIX paeHa t. Ho Hae Mo>KeT HHTepecoear& 1-IHCJIO N 1 (t) coo6meHHt\, ,ll.JIHHa KOTOpbIX MeH&we l1JIH panHa l1MeSl B Bl1.lJ.y,
t.
41
59
OBCYiKJlEHHB
qTo
t
Beer JI.a ueJ1oe, HMeeM: s=f
Nt(t)=
!
s=t k=cr
N(s)=
! l
s
s=I
s=t
k=':!
AkXk=
1 k=I
!
Au
k=l
!xi.
(-1.19)
s=I
==
cyMMY cnaraeMoe s 0, TaK KaK coo6rn.eHHe HyJJeBof.t ,lJ.JlHHbl He HMeer C~lhICJJa. l1cnoJib3YH COOTHOWeH11e
Mhl He BK ..1I04HJIH
B
nonyllaeM:
(4.21) rJI.e
1-x; Mhl MO)l{eM nepen11caTb :rror peayJJhTar cne.lly10rn.11M o6pa30\'1:
( 4~22) rJJ.e k =cr
-· ~ B o/ B k•
.....
k
LJ11cJio N 1 coo5w.eHnf.t, Jvup-1a KOTOph1x MeHhllie 1.J:eM t, HJIH pae: Ha t, co.J.ep>KHT npeo6.11ana10m11fi 4JJeH c KO.Tie6areJJhHb1e '-lo11eHbl (k = 2, 3, ... , a) H IlOCTOHHliyIO Bo. OropaChlBaH Kone .. OaTeJihHbie 1.J:JieHbl, l'fMeel\I:
Xf,
(4.28)
60
IlPHHUHO~
JVH1 OOJlbllJHX
t.
fr.rr. 4
K04HPOBAHHH
Mbl no.rryr.iaeM, TaKHM oopa30M,
l l n N~ +Bo = t o t = -~81
+1
-~-In
(N1
1
+B
( 4.24)
0),
r.n.e
to== -
-rr1 ln B,.
5. fipHMepbl TioKameM .n.eACTBHe Meroaa Ha HeCKOJlbKHX npMMepax.
A. B c e B
c H M B o JI hI
H M e 10 r
o .n H H a K o s y 10
=
9TOM c.rry4ae MhI 6epeM a 1 n -- o6meMy JIOB .nJIHHhl t 1 = o 1. PaeeHCTBO ( 4.12) .naeT:
=
a JI H H y
r.i11c.11y CHMBO ..
(4.25)
X-n=O. H:\teeTcsi OJl.HH BemecreeHHhifi H CJleJ{OBaTeJlbHO, t
N
110J1om11reJ1bHhiti KopeHh
,,
B.
C H M B o JI bl
B. 1.
B
CTeneuh 061u,eM CJ1yr.iae
•
n
II B y x
= n,
nt -- l
=n uN I == 1-n i --1
9TOT KaHaJI 06J1aJl,aeT a6COJIIOTHO.i:t JIH6o KOJie6aHH.i:t.
X
ycrottr.iHBOCTbIO 6ea K8KHX~
p a 3 JI H r.i H hi x
xapaKTepHCTH 11ecKoro
l
.n JI H H:
ypasueHH51
H a paeHa
o.
( 4.26)
B. 2.
TipocTeAun1e cooTHollleHHH noJ1yLiaeM npH 2 )( -
a 1X- a 2
o = 2:
= o.
(4.27)
KppnH aroro ypauueHH~ paBHhl
( 4.28) T:a~K. !opMy no" (4.15), a eCJIH KO)lHpoeaHHe ycro"llHBO, Mbl MO)l(eM 80CIIOJib30BaTbCSI ynpomeHHblM BbtpalKeHueM ( 4.18). 4HCJIO CJIOB 11eHOD MeH&llle t HJIH pasHo" t Bblpa)l(aeTcst senuq11Hott N 1 (t), onpelleJIHeMott no Q:>opMy ne ( 4. 22), KOTopaH )lJISl YCTO"'UtBOfO KO,lla npHHHMaer ynpome1nthtt1 BHJJ.
( 4.23) PacnonoiKHM Tenepb see cnosa cnoeapst s nopst11Ke noapacTaHHff u.eHLI t. t5y ACT oaHat.1aTh rropstJJ.KOBbl" HOMep cnosa B 9TOM CIIHCKe. B cpellHeM COOTHOllle1111e Me)f(JlY u.eHO" t M noMepoM BMpaiKaeTCH KaK
N;
N;
t
= t 0 + ~- In (N; + 8 0).
( 4.24)
PaccMorp11M Terreph qacToTy (11n11 sepostTHOCTb) cnoea. Me)f{llY u.eHoH 11 sepoSITHOCThlO Mor no Ohl 11 He cyw.ecTeoaarb HHKBKO" CBH3H, 0,[(H8KO Mbl npe,llTIOJIO)l(HM, qyo nepoHTHOCl'b cnoaa 11 ero ueHa Ha)lJie)l(aIUHM ot5paaoM cor nacosaHhI. 9To oana .. 4aeT, 4TO pe,llKHe CJIOBa HMeIOl' JlJIHHHoe llOporoCTOSILUee KOllOsoe ot5oaHa 1-1en11e, a Ha11t5onee ynoTpet5uren&Hble cJioBa -- KopoTKoe H 3KOHOMHOe KOllOBOe 0003HaqeHHe npot5neMa cor nacosan11s:i t5b1na ot5cy)l(JleHa s nrr. 8 u 9; t5b1JI0 BblHCHeno, KaK c.11e,llyeT BblOpaTb HaHJIYt.IWHM ot5pa30M sepoHT· HOCTH CHMBOJIOB, HMeIOW,HX pa3JII14Hhle ueHhl tj. Peay nb Ta T Bh! .. pa)l{aerc\-1 cpopMy no": HJIH
-- 'j
PJ = e
·1
t
i.
{ 4.61)
flpuMeHHM -renepb CXO)lHhte paccy>KlleHHH He K C.HMBOJiaM M.'111 t5yKBaM, a K CJIOBa!\I. 'IOBO aa HOMepoM HMeeT u.eHy t N'' rrp11qeM o6e seJIHQHHbl CBSl3aHbI COOTHOllleHHSIMH ( 4.23)
N;
c.
1
11 ( 4.24). Hawe nosoe ycnosue xopowero cor nacosaHHst co· CTOHT B TOM, llT06hl eepOstTHOCTb p N, CJIOBa OblJia CBH3aHa c 1
ero u.eHo" tN' ot5m11M ycnosnel\1 1
(4.61), co11epiKalllHM Hosyio
npo11asoJI&Hy10 nocTostHHy10 ~ 1 • Conocrasnsisi aru Q:>opMyJihl, Mbl MO>KeM HCJ(JllOt.IMTb ueHy tx· H CDSl38Tb nopSl)J.KOBblii HOMep t
N;
74
·[rn. 4
TIPHlflUlllbl KOJlJ1POBAHHSI
cnoaa HenocpeacTseHHO c
~ro
eepoRTHOCTb~ PN~
(4.69)
(4. 70)
=
r JI.e P e- ~ito, 1 = ~ 1 /~. CToHHTHe o ~ Ko,.u,11posa111111 cnos a Mosry H 0 ueHe K040BYX o6osHaqeHH~ DOTpe6oBaJIOCb ~Jiff BblBO.lla COOTHOWeHHit ( 4.23) H ( 4.24), HO Mbl nocTapaJIHCb HC• Kn10qMTb ne113secTHh1e ueHhI, npe4noJIO)l(HB Ha4neiKamee cor JiacoBa1111e. opMyna ( 4.70) Bhlpa)l(aeT saKon, npeJIJIO)l(eHHhtit MaH11en&· 6poToM. OH BKJI10t1aeT, KaK qacTHblh cnyqa~, aaKoH, BbICKa3aHHhiit UuncpoM:
PN;=_~;• P".'"'·1~.
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(~1=~).
(4.71)
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a cyMMa sepoSITHOCTeh ,llOJI)I( .. Ha paBHSIThCSI eAHHHue: N'1 = R
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paseHcTaa
(6.1 ). B Tat5JIHUe 6.1
Jl,aHhl HaH60JibUJHe llHCJJa CHMBOJIOB, KOTOpbie Moryt t5MTb npoBepeuhl npH pa3JJHqHblX 3HalfeHHSIX k.
T a 6 n H Jl a 6. l Han60JibWee lfHCJIO CHMBOJIOB, KOTOpble MOfYT 6bITb npoBepeHhl npHMeHeHHeM OT OJl.HOfO Jl.O DSITH npoeepotJHblX CFIMBOJIOB lf ncJto npouepoqnblX Bcero CHMBOJIOB
k
CHMBOJIOB
n
H nop Ma1.u1onllblX en M BOJIO B
m=k-n
I 2 3 4
l 3 7 15
4 II
5
31
26
0
1
BTopott cJiyqati e Tat5Jiuue 6.1 cooTeercrnyeT npeJ1.HOJIO)l{eHl1IO, llTO B nocJiel{OBaTeJlbHOCTH 113 tpex JlBOHqHblX IUHpp HliKOr JI.a He MO)l{eT 6h1Tb t50Jibllle, 4eM OJI.Ha OlliH.6Ka. 0l{Ha H3 9THX UHcpp CJIY:iKHT JlJISl nepeJlalfH HHcpOpMaUHH, a JI.Be J1.pyr11e J1.JIS1 npoeepKH, cor11acuo ta6JI11ue 6.2. ITpoaepo4Hbie u11cppb1 B 11 C Bh1611pa10TcS1 TaK, qro6b1 (A+ B) 11 (A+ C) 6b1JI11 qeTHhlM11. ITpHeMHliK BblllHCJI5IeT o6e cyMMbI, 11 eCJIH 0.U.Ha 113 HliX OKaSb1BaeTC5I neqeTHOit, npoaep04HhlH ClirHaJI B HeH Collep:iK11T olll116Ky. EcJI11, Hanpn:Mep, (A+ B) He4eTHa, Tor lla KaK (A+ C) qeTHa, To curHaJI B Heeepeu. EcJIH HMeeTC51 om11.6Ka B cHrHa11e A, To o6e cyMMhl 6y JI.YT HelfeTHbIMFI.. OtcroJ1.a HcHo, KaKott c11rHaJI JlOJIIKeH 6h1Tb ucnpaeJieH. Tper11tt cJJyqatt - KOJl «tieTbtpe 113 ceMH»: tieTb1pe .U.BOlilfHhIX uucpphl JlJISl liHcpOpMaUHH, Tp11 JlJISl npoeepKH. 0lIJH6KH Moryr COOTBCTCTBYCT pa3.'ll141110 B OJlHOM llBOWlJHOM 3HaKe, a paCCTORHlie k enwBHU-pa3.'1HtUllO B k ns:JHllHblX JHaKax. TaK. HanpHMep. onacaHHoe B n. I Jlo6asneHHe OJlHoro nposepolJHoro 3HaKa (0 HRH I) ysenwmsaet ua eJlliHHUY paCCTOSIHHe MeiK.ltY pa3peweHHblMM KOM6HHaUHRMH (o6na,IlaIOll\HMH lleTHblM 'IHCno.M e.a;HHHU). (npuM. nepea.)
96
KO.ll.bt, OBHAPY1KHBAIO.UU1E H 11CUPABJISIIOIURB OlllH.SKH
lr Jl.
6
6h1Tb
o6ttapyiKeHbt H JIOKaJIH30BaHbl
4TO
a
B
npe.LI.lfOJIO»CeHHH,
rpynne H3 ceat11 .a.eonlfHhlX u11cpp HHKor.a.a He ecrpettaeTcsi 6onee
T a 6 n H u a 6.2 CxeMa KoJJ.a c o.n.uoA u.11cf>poA AJIH uncf>opMau.uu u ABYMH JJ.JIH npoaepKH Ha¢>oparau1u1
OpoaepKa
A
B C
x
x
x
0 p H M e q a H H e. JlOJI)KHbJ 6blTb Bhl6paHbJ
x
flposepKH raK, qro5bl
«X> no K3)KJl0'1 crpoKe nasana qerHoe qHcno. CYMM3 Cl-IMBOJIOB
OJI.HOA OlllH6K11. fipoeepKa SaHO B T aOJUIUe 6.3.
MO)l{eT
ocymeCTBJIHTbCH,
KaK
noKa·
T a 6 JI u u a 6.a
.l{oJI. c qeTblpbMH .ll.BOH'IHhlMH UHcf>paMH AJIH HHcf>opMallHH H TpeMH AJIH npoeepKH Ma4>BKTRBHOCTh RCnPABJIHIOIUHX KO)lOB
99
C.
D.noRHaSI OlllHOK8. nocJieJI.HSISI npoaepKa .n.aer lfeT, a He• KoTophle H3 npe.ll.hlJI.YlllHX - He11eT, llTO yKashlsaer Ha HMe10weec~ r )le-To HCKa)l(eHHe. MeTo.a RBJIHeTcst 06m11M H MO)l(et OhlTb npHMeHeH K JllOOOMy npHMepy, OOCY)l(JI.eHHOMY B npt:.ll.MllYUlHX pasne.r1ax. X9MMHHr yKasaJI reoMerpHlfecKy10 HHTepnper auHto, npnMeHHIOIUYIO npe)lCTaBJieHHe B MHOfOMepHOM npocTpaHCTBe. 8ro npe.n.cTaoJieHHe J1.aeT a.r1eraHTHoe .n.oKaaarenbcroo OOIUHOCTn onHCaHHblX MeTO.ll.08. Hoahle uccne)loeaHHH no TeopnH ucnpaBJIS110IUnX KO.LI.OB coJI.ep)l(aTcH o pa6ore ,UJKn.r16epr a 1). 06IllnpHbil1 MarepHaJI no KO· .n.a~I, HCnpaBJISllOIUHM MHOfOKpaTHble OIIIHOKH, MO)l(HO Hattrn B Trans. IRE 11)
4. 34>4>eKTHBHOCTb ucnpaBJIHIOIJlHX KOJI.OB HcnpaeJIS110IUHe KO.llbl, onHCSHHhle B npeJ1.hlJ1.y111ux paaAenax, MoryT 6hITh ncnonhsoaaHhl c.r1eny10m,nM oopasoM. BoahMeM e Kaqecree npHMepa KO.ll. «4 H3 7» (CM. ra6nnuy 6.3). HaqHeM c HeKOTo· poro COOOll.leHHSI, KO.ll.HpOB3HHOf0 .ll.BOH'IHhlMH UHQJpaMn, HanpHMep:
(6.5)
0000100010100100 u paaJ1.enuM ero Ha rpynnhl no lfethlpe UHQlphl a Ka)l(.n.oit:
0000,
1000,
1010,
0100.
(6.6)
,UonoJIHHM renepb KaiKJI.yIO 113 9THX qeTblpexsHalfHhlX rpynn erue TpeMSI SHaKaMH, COOTBeTCTBylOIUHMn npoeepKaM na lfeTHOCTb, cor.r1acHo (6.2):
0000000,
1000111,
1010010,
0100110.
(6.7)
8To coo6w.eHne nepeJI.aeTcH, npHeMHHK npwMeHsteT npouepKH Ha qerHOCTb ltJISI HCnpaoJieHHSI OlllHOOK, 3aTeM ycTpaHSleT HSJIHillHHe nocneJ1.nne rpu snaKa B Ka)l(.n.oit ceMH3Ha'IHOD rpynne u eoc· CTaHaBJIHBaeT HCXO,llHOe coooruenHe B BH.ll.e (6.6) H (6.5). 3TOT MeTO.lJ. rOltHTCSI 8 TOM c.r1y11ae, ecJIH Ha n CHMBOJIOB HnKor JI.a He npuxo.n.Hrcs:i 6onhrne onHott ornHOKH (a HameM npH· Mepe n = 7). 3to ycJIOBHe HHKOrJI.a B TOtIHOCTH He BhinOJIHSJeTCSI .. 06bllfHO H.'.\feercH COBOKynHOCTb CJiyqattHblX ornu6oK co cpe.ll.HeD uepo11rHoCTh10 p. MhI noKaJKeM, llTO HCnpaaJistlOWHtt KO.LI. MOJKeT
E. N. 0 i I be rt, BSTJ 31, 504 (1952). Trans. IRE (PGIT) 1-4, CM. a oco6eHHOCTH e r. 4 crat&H 0 o I a y, E 1i a s, R e e d, S i I v e r m a n, Ba J s er u Jlp. CM. taK>Ke Conv. R.ec. IRB National Convention, 1954. 1) 1)
4•
100
KOJJ.bl, OBHAPYIKHBAIOIIlHB H HCflPABJISIIOUUIB OWMBKH [ r JI.
8Ha4HTeJihHO yMeHhWHTb lfMCJIO OIIIH6oK M .n.aTb HOCTb p HencnpaBJieHHhtX OWM60K.
p
IlycTb
03Hal.faer
(k, n, m),
B.>3bMeM KOJJ.
m
HHcpopM3UHOHHblMH
n
llBOMlfHhlX
0
nepexoJI.
1
a
1
HJIH
0.
e
T. e. KOJJ. c
k
npoaepolfHblMM 3HaKaMH
IlOSHUHflMH,
a
ocero
6.1. B
H03HUMflMH, cor JiaCHo ra6n11ue HOCTM 113
MaJiyIO oepOSIT·
03H34BeT oepOSITHOCTb OWH6KH B HeKOTOpOM onpe·
.n.eneHHOM BHaKe. 3ro H
6
u11cpp
HMeeM -
n
c
.naHnoli
llBOH4.HblMH
nocne.n.osareJib·
CJie.n.y10w.11e
oepostTHOCTM:
eepOSITHOCTb otcyTCTBMSI own6KH;
-
aepOSITHOCTb 0.Ll.HOli OlllM6KH;
-
(6.8)
own~
eepOSlTHOCTb JJ.BYX
6oK a JJ.Byx pa3HbtX no3HUHstx;
nl
1tz =ti (n -
l)!
l
P
(l
-
)n-l
- P
PaccMorpHM CJiy11atl
l
B OllHOit
B
nonyqaJOrcg aeJI11tIHHht 1tz.
orn116K11,
9TOM
6oli ns
p),
owu601
Keune nesepHhlX snaKoe. EcnH neeepttbllt 3HaK 6hu1 npHHHT KaK 1, TO ero ttymtto npoc10 aaMeHHTh Ha 0, H uao6opo1.. MHct>opMaUHH o noJ1omeHHH OWHOOK nepe.n:.aeTCSI ny1eM .aooaeJJeHHff "neu11aJibHblX CHrHaJIOB K .llBOH4HOMY KO.llY (0 .llJIH npaBHJlbHOfO, l .llJlff ueeepttoro) c eepoHTHOCTblO p ,ll,JIH l H (1 - p) .llJlff 0. llJJH 9THX .ao6aBOlfHblX CHrHaJIOB Tpe6yeTCH KaHaJJ c nponycKuon cnocoonoc1h10
-plog9 p-(l -p)logi(l -p) .lI.BOHqHblX e.llHHHU Ha CHMBOJI. 3Ta ,lI,OTIOJIHHTeJI&HaH HHcpOpMaUHR, oy .ayq11 nepe.aaeaeMa no TOMY JKe Kanany, yMeHbIIJaeT ero nponyCKHYIO CTIOCOOHOCTh. l1CXO.llHbllt KaHaJI 6e3 lllyMa HMeeT 1, a KaHaJI c wyMoM HMeeT, cne.aoea1eJibHO, nponycKtty10 cnocooHOCTh, sh1pa:iKaeMy10 cpopMynolt (6.12). Teope1HqecKH aoaMOiKHO HattTH MeTO.ll. KO.ll.HpOBaHHH, peaJJH3YIOillHll TaKyIO nponycKHYIO CnOCOOHOCTh.
c=
') C. E. S ha n non, W. \Ve av er, The Mathematical Theory of Communication, p. 38, U. of Illinois Press, Urbana. 1949. (PyccKnti nepeuo.It B c6opHHKe cTeopHfl nepe.naqH 9JieKrpHtieCKHX CHfHaJIOB npH HaJIHIJHH noMex>., HJI, MocKaa, 1953.)
5)
DPODYCKHASI CIIOCOBHOCTb .llBOMllHOfO KAHA ..'IA C lllYMOM
103
o-'l
npe.nbl.nymeM npMMepe npn p = 1 cpopMy na (6.12) .naeT nponycKHYIO CilOCOOHOCTb OKOJIO 0,92, 4TO 03HattaeT, llTO TOJibKO OJlHH .llOIIOJIHHTeJlbHbllt .llBOHllHbllt 3HaK Ha Ka:tK.llble .neB71Tb ncxo.ntthlx attaKOB 6bIJI Obi .nocTaTotteH JJ.J111 ucnpasJieHMlt. Ha1u
B naweM
KO.ll «4 H3 7» He .naeT IlOJIHOfO HcnpaBJieHHfl H TpeoyeT 7 3HaKOB Ha 4 HCXO.llHblX 3HaKa, llTO .naeT yseJIH4eHMe Ha BMeCTO 1 Mbt ottettb .naneKH OT TeopeTH 4eCKoro npe.nena. LliKHJ16epT npHXOJlHT K TaKoMy iKe aaKJllOlleHHIO, He npeJ(JiaraR HMKaKoro
3/4
/9.
npaKTM11ecKoro pemeHM11. Ero cTaTbfl co.nepiKHT HHTepecttoe oocyJK.neHMe CBOllCTB OOJib rnoro llMCJia pa3JIM llHblX KO.llOB. TipooJieMa .llJIR CB713M,
MCnpaBJISllOlllMX
M aa nocne.nttee
KO.llOB BpeMH
HMeeT 60JibIIJOe 3Ha11eHMe
9TOft
11po6JieMe IIOCBflIUeHo
Mttoro pa6oT. MeTo.n X9!\fMHttra (CM. pa3.nen
2)
coa.naH .nnsi asTo·
M8TH11ecKoro ncnpanneHM" Heeepttoro cnMsona. 6011ee 9KOHOMH4Hblft
MeTO.ll
OblJI
rrpe.nnoiKeH
COCTOMT B uaxom11eHHM
BarttepoM
CHMBOJf 8,
(MIT).
Ero
npnHunn
Henepuoro c HaMOOJlblllelt Be-
9TOrO CHMBOJla. 3TOT MeTO.ll 1 06cym11ett CnnsepM9HOM " BoncepoM ) H npeJJ.CTaBJ1ReTc11 MHoroooema10mMM. llJIH OOHapy1KeHH71 CHMBOJia, KOTOpbllt MMeeT HaMOOJibWYIO eepoHTHOCTb 6hITb HesepHbIM, asTOpbl HCIIOJih3YIOT
pOHTHOCTbIO,
M B
HCnpaeJieHHH
HeKOTOpble anpHopHhle JlaHHbie o naaH~OCBHa11x nepe.naeaeMoro cooomettn11. HaJIM4He B3aHMocs113ef.1 oaHattaeT M30bIT011HOCTb M yKa3bIBaeT, l!TO llMCJIO npMMett11eMblX .llJlfl nepe.na4H CMMBOJIOB cnemKoM eennKo. Bb1QMCJieHM11 11ncna Heooxo.nMMblX CMMnonon, BhtnonueHHhle CHneepM3HOM " BoJicepoM .n~H HX MeTo.na H Me· ro.na X9MMMHra, no-en.nn~oMy, He ytti-1Tb1ea10T 9Toro oocTOHTeJibCTea. 011HaKo OCHOBHafl n.nesi npe.ncTaBJISleT OOJiblllOlt MHTepec, M K OiKM.llaeMoll J18Jlbttetlweft .llMCKyCCHH 0 pa3JIM11HblX MeTo.rr.ax HCnpaBJieHHSI cne~yeT OTHeCTHCb co BHHMaHHeM.
Silverman, Balser, PIRE 42, 1428 (1954); Trans. IRE (POIT) 4, 50 (1954). (PyccKHit nepeaoJt a c6opHHKe , HJI, 1956.) 1)
r .rt A BA
7
IlPl1JIOJKEH11SI K HEKOTOPbIM CilEUHAJlbHbIM IlP06JIEMAM 3aa;a11a 0 paCDOJIOlKeHHH c DpHMeHeHHeM CMemaHHOH HlJeJiKH
J.
EcnH Mbl >KenaeM KnaccHcpHUHpoeaTh JJ.aHHhie (6yKnh1, ornH6KH, pacTeHH~ H T. Jl. ), pacnOJIO)KHB HX B onpeJJ.eJieHHOM nopHJJ.Ke no Ha6opy 51qeeK, TO B03HHKaeT BOnpoc 06 OIITHMaJibHOM pasMepe st11ei!KH. EcnH SILieltKa MaJia no cpaetteHHlO c pa3MepaMH 3neMeHTOB, TO BOSHHKaeT ueonpen.eneHHOCTb; eCJIH SllleUKa ClIHWKOM BeJIHKa, TO pacnono>KeHHe oy n.eT ne3eKTHBHblM. PaCCMOTpHM 9TY 3a .. n.a11y H HCCJien.yeM pOJib pa3JIHlf HbIX B03MO)l{HblX npaBHJI pacnoJIOiKeHHSl. Mb1 MomeM npHMeHHTb c.MemaHHy10 st11ettKy ( miscel· !aneozzs cell) ll.JISI 9JieMeHTOB, npHHall.Jie)KHOCTb KOTOpbIX K KaKO:tt• JIH6o onpen.eneHHOlt ~qettKe HeSICHa; Mbl MOJKeM TaK)Ke npHMeHHTb CHCTeMy nepeupecmHblX ccbl1tou (cross-references). 11HTepecuo cpaBHHT& eeKTHBHOCTb ooonx MeTon.oe 1). Hawe paccymlleHHe OCHOBbIBaeTCSI Ha oowett cpopMy ne ( 1.13 ):
l=-K
_!p1 lnp1,
(7.1)
J ablpaiKa10u1e1t HucpopMaunro ua CHl\IBOJI n.nsi cnyqa51, Korn.a JtenaeTCSI e1>1oop HS ueKoToporo 11Hcna cooblTHlt a, b, ... ,j, ..... , HMerowux anpHOpHble BepoHTHOCTH Pa,pb, ••• , p1, ••• flpH npH· MeHeHHH ll.BOHllHOii cecTeMbl K = logg e,
I=-
!.v 1og p 1
2
1,
(7.la)
i H HHcpopMaUHSI HSMepsieTCSI B ~BOHllHblX ell.HHHU3X. 1)
D. K.
3ra npo6neMa snepeble nocraeJICHa 11 o6cyiKJleHa MaK-.l(oHaJILJloM: C. MacDonald, J. Appl. Phys. 23, 529 (1952).
105
t] B KaqeCTBe ynpomeHHOH oTpeaKa JVIHHOH l Ha JIH1u11'l L TaK qTO KpaeshlM1'1 Jl1n-111H L paaJJ.eJieHa na N
cxeMbl paccMoTpHM pac·noJioJKeH1'1e noJIHOH ll.JI1'1Hhl L; noJIOJK1'1M, tJTO 9cJ>cpeKTaMH MOJKHO npene6peqb. cMeJKHbtx cerMeHTOB D.JIHHOH m,
> /,
TaK 4TO
(7.2)
Nm=L.
llJI1'1Ha m npe.a.cTaBJIHeT paaMep srqetiKH, " Mbl HmeM OflT1'1MaJibHYIO D.JIHHY m, T. e. TaKoe ee 3Ha1..1ettHe, Koropoe n.aeT HaH60JihlllYIO HHrpop:v.1auH10. 3ra cxeMa npeJJ.cTaaJieHa Ha pwc. 7 .1. PacnonoJK1'1M CJiyqaBHhtM o6pa30M Ha JIHH1'1H L cerMeHT AB D.JIHHOti I. Orpe30K CD npen.cTasJIHeT on.uv H3 siqeeK D.JIHnott m. Ecn" TOqKa A
m
! ...
..1 ....
I I
a-L-'
I
I
I
c
,..
I
I
A
I
J
Ill
I
I
I•
I
...
.,
I
L I
l
I
[
lJ
I I
I
m-l
....,
Puc. 7. L M o.!l,e JJh pa cnonomeHHH c Hqe.ltKaMH Jt.:tHHoA m =CD H IlJJMHotl pacno.'IaraeMbIX 9JieMeuros I= AB.
nona11aeT Me>KJJ.Y C a E, TO cerMeHT AB noJIHOCThto ttaXO.ll1'1TCH snyTP" ff qei1K1'1. EcJIH A nonan.aeT Memn.y E H D, TO cerMeHT nepeKpbtaaeT TOllKY D - MeCTO CThtKa n.syx HqeeK. Mbt HMeeM:
CD=m,
ED=!,
CE=m-l.
(7.3)
Ilpen.nonaraeTCH, qTo no11anan11e AB a J1106y10 aqeHKy eMeeT onaHaKosy10 sepoSITHOCTb. BepoSITHOCTh roro,. 4TO AB oKamerca noJIHOCTbto BHYTPH i-H SilfetiKH, ecTh
m-l
Pi= L
(7.4)
m
Pi= I JlJISI TOllellHblX 9JieMettToe). llTO AB oKa>KeTCSI nonnocTbro BHYTP"
(910 BhlpaJKettae CBOD.HTCSI K
IloJIHa51 eeposiTHOCTh Toro, Ka K 0 tt ... JI" 6 0 51qei1KH, paBtta i=N
P= ~ m-1 ~ i=J
L
= NopMaUHH, nonyqaeMoe
I/
B pesyJI&TaTe rrepeKpeCTHblX CCblJIOK,
In (2ml-/)
rerrepb, KOHe'IHO,
uttoe; naxo.a.R MaKcnMyM I no m, n, nonh3YHCb npe>KHUM 060anaqeuneM
m
7
= x,
nonyttHM ycnoene onTHMyMa; X=
2x-l
2x+l
2x-1 x-1'
ln--
(7.11)
108
fiPHJIOiKBHIUI K HBKOTOPblM CfiBUHAJlbHblM fl.POBJIRMAM (r.'I.
qTo llaeT x ~ 3ae11cnr OT L.
1,06;
7
HHrepeCHO 3aMeTHTb, qTo 9TO 3HalfeHHe He
B
TOM JKe cnyLiae, KOrlla Hawa KJiaCCHcpHKaUHOHHaH CHCTeMa CJIY)l(HT llJIH Henocpe.rr.cTBeHHOro HaKonneHHSI CTaTHCTHlleCKOit HHcj:>OpMaUHH, nepeKphlTHe TI03BOJIHeT HaM JIOKaJIH30BaTh 9JieMeHT. 8TO npHBOllHT K COOTHOllleHHIO
I)
I [( m- l l l] K = - 1 -m- ln l +-lnm l ' H ycnosHe onTnMyMa
6ylleT
(7.12)
Tenepb
x=-ln(x-1),
(7 .13)
OTKylla x ~ 1,28 - 3Ha4eHtte, TaKJKe He 3aancsi1uee OT L. CpaBHHM Tenepb KonnqecTso HHcflopMaunn Ha 9JieMeHT .rr.nH pa3J1HqHhlX MeTOllOB pacnOJIOJKeHHSI, BCHKH:tt pa3 npH OilTHMaJibHblX ycnoBHHX. B nepaoM cnyqae llBOtlHoro pacnonoJKeHHS1 npn onTHMaJihHOM 3H34eHHH ni/l Mbl HMeeM:
+ l n ( m-
I ) l l ( K opt= " l
I
) l -
+
2m l 2m - l'
(7.14)
HJIH
(k )opt= In ( ~) = 0,03,
(7. l 4a)
Tor4a KaK BO BTOpOM cnyqae
(k )opt= In ~- + 0,28.
(7 .15)
CpaaneuHe 9Toro nocnellHero 3HaqeHHS1 co cnyt1aeM cMemaHHoii ffqe:ttKH, paCCMOTpeHHO:tt B paa.rr.ene 1, noKa3blBaeT, 4TO CHCTeMa 4aottHoro pacnono>KeHHH oKa3hIBaeTCH nyqwei:t npn ycnosHH
m-1 1 T. e. Kor4a
+ In m-l / ;;:::: -
L
1, 11,
l
a npH TaKoM ManoM 3naqeHHH
L
CMhlCJia. 3aMeTHM
QTO
s
3aKJIIOQeHHe,
0,28, (7.16)
3allaqa pacnonoJKeHHH nuwaeTCSI He npellCT asnsieT rpy .rr.a pac...
CMOTpeHHe cxeMhl, B KOTOpott 9Jiel\leHTbl J(aHHblX HMetOT HeKOTopoe
3]
HAHBNf04HBRWBB qncno 9nBMBHTOB HA 04HY ~qBttKY
109
cTaTHCTHqecKoe pacnpeaenenne no BJIHHe. TaK, ecnn nnoTHOCTb eepoaTHOCTefi l ecTh p (l), TO, HanpHMep, an51 pacnonoJKeHHff co CMernaHHOR srqeRKott (CM. (7. 7)) HMeeM:
k=-5 {(1-~)1n[(1-!)[J+·!ln~}p(l)dl. (7.17) 00
EcnH B3RTh
p(l)={
i
DpH
Q~ {~lo,
npn
t> 1
0,
TO nocne HeKoTopwx npeoopaaosaHutt rronyqaeM ycnoeHe MaKCH· ManhHOtt HHcpopMaUHH:
m_ 1 lo -
+1n (
L
qTo .llnSI cnyqa51 ·-lo
=e, ri
Lio
)
m (m - lo) -
)g
(m 1 ( m ) lo n 11l - lo '
paccMoTpeHHoro paHee, .llSeT: m lo= 2,2.
(7 .18)
Bo ecex paccMOTpeHHblX npnMepax Mbl nMenn .lleno c 9neMeHTaMH, KOTOphte MoryT ObITh KnaccncpnunpoeaHbl (pacnonoJKeHbt) eaonh JIHHHH, B OBHOM H3Mepettnn. Cxo.nHhte coo6paJKeHH51 MoryT ObITh pa3BHThl anH 9JieMeHTOB, BaBHCflllJ.HX OT JlByx, Tpex n 6onee nepeMeHHhlx; TaKHe 9neMeHTht BOJIJKHbl ObJTh KnaccHcpnuHposaHhl Ha IIJIOCKOCTH, B o6oeMe HJIH B MHOfOMepHOM npocTpaHCTee. B 9THX ycnosHSIX cymecTBeHHO H3MeHHTCR nono>KeHHe c nepeKpbITHeM, TaK KaK OBHH 3JieMeHT MOJKeT Tenepb nepeKpbIBaTb HeCKOJihKO pa3JIHQHblX RqeeK.
3. Haee&1ron.neuwee 11ucJ10 9JieMenToe na on.uy sJieMenTapny10 H11eifKy Ha sonpoc o Hans1>1roanettweM qncne 9..1eMeHTOB MO>KHO nonyqHTh paanHQH~e OTBeTw s saBHCHMOCTH OT noaxoaa. CTaHeM Ha TOQKy apeHHR ¢naHKH H, paccMaTpHeaH uenu c onpeaeneH· HblMH IlOCTOHHHblMH BpeMeHH, 6y.neM CTpeMHTbC51 rronyqHTb Fianoon&IIIyto 9cpcpeKTHBHOCTh. H11mecneay1.0w_ee paccym.lleune
110
DPHJlO>KBHIUI K HBKOTOPLIM CnBUHAJlbHhlM nPOBJIBMAM [ r n.. 7
npHHa.nne>KHT P. M. YKHM, qTQ Mhl ucnoJih3yeM n paBHOOTCTOHIUHX 9HepreTH4.eCKHX ypOBHeft
(7.20) Ecnu eo36y>K.lleH uauehlcwun ypoaeHh En u 3aTyxattue npoucxollHT cor nacno (7 .19), TO, oqeeun.Ho, Mhl .llOJI>KHhl JKJJ.aTh, noKa E ynaJJ.eT llO 3Hat1eHHfl HH)l{e £ 0 rrpe1Klle qeM Mbl CMO>KeM OTC4H· TaTh cne.ny10my10 9Heprerut1ecKy10 cTyrrettb, Koropas MOJKeT 6h1Tb 0 unu E0• 3To eelleT K ycnoeu10
/2,
~=(n-l)Eoe-tl•, t
~=
I
- ln 2 (n _ l) = In [2 (n - 1)].
(7 .21)
BpeMs t, onpeJJ.enseMoe paeencTBOM (7 .21 ), npeJJ.cTaBn51eT KpaTqaRmun HHTepean Me>KJJ.Y JJ.BYMH cne.ny10muMu .llpyr 3a .npyroM HMny nhCaMu, ecnu Mbt XOTHM paanuqaTh HMnynbChl Tuna (7 .20). Ha npoTHJKettuu 6onhllloro apeMeuu T Mbl nMeeM oomee qnCJIO CMMBOJIOB
Q-I_-
t -
't
T ln [2 (n - 1)] •
(7.22)
nonaraH, qTo nee HMnynhchl HMe10T OllHHaKOBble anpuoptthte BepoHTHOCTH, nonyqHM HaMOOJiblllee BOSMO>KHOe KOJiuqecTBO HHcpOpMaUHH (CM. (1.6)):
I= KO ln n = KT f(n), 't
(7.23)
rAe
f
Inn
(n)
= ln [2 (n _.:.. l)} ~
(7.24)
3]
HAHBYrOnHBftmEE qncno 9nEMBHTOB HA OJI.HY HQEDKY
111
f
nocTpOHTL rpacpnK (n) KaK cpyHKUHIO uenoro 11nc.11a n, HaMeHHIOWerocsi OT 2 (JJ.BOH'IHblit KOJJ.) JJ.0 00 ( HenpepbIBHOe naMeHeHne); 9TOT rpaKno
f
'
I
0,77 -·· -
~6
0
*
...___,_--'------------n·
__.......___....,_....__.__,__._..__.-.a-......__
I
2
J
Pee. 7.2. rpacpHK
n = 4,6
I
5
(n)
6
7 8
9
= (ln n) fln [2 (n -
IO
1)J B cpyHKUHH n.
H, B03pacTaSI, CTpeMHTCff CHOBa K eJJ.HHHUe npH 011en& 6onhWMX 3Ha'leHH51X n. 3THM J10K338HO, QTO KOJIH'leCTBO HHcpOpMaUHH, nepeJJ.aeaeMoe aa .ll8HHOe epeMff, MaKCHManbHO B cny11a5IX JJ.BOH'IHOro TenerpacflHOro KOJJ.a H TeneciJOHHOD CBSI3H c HenpepNBHWM H3MeHeHHeM.
fJIABA 8
AHAnH3 CHfHAROB:METOn ~YPbE H IlPOUE,llYPhl OTCqETOB 1. Psu1 yphe B
nocJieJiyromnx rJiasax HaM noHa.n.oosircsi HeKoropr.1e MareMBTHlleCKMe pe3yJibT3Thl, OTHOCHUI.MeCH K aHaJIH3y pa3JIHqHoro po.lla CHrHaJIOB. H acToHmaR r JiaBa nocaHIUeHa c11creMaT11qecK0My 0030py MaTeMaTMqeCKHX MeTOJIOB H HX B33HMOOTHOllleHHft HallHeM TiycTb / (t) UMSl MO>KeT JISlIOIUHX. B
c pasJio):ofdt,
c_,.= c,.*,
(8.20)
- i:;2
f
r.ae (t)- eeruec1eeHHaff cpyHKUHH. TaK KaK HMnynhc paeeH Hyn10 BHe KOpOTKOrO HHTepeana 6, TO npe11.eJJh1 HHTerpHpoBaHHSI MO• ryr 6b1Tb B3SITbl paBHblMH -1- 6/2. Tipe.ZinOJIO>KHM, l{TO paCCTOSI• HHe 't Mem11.y coce.aHHMH HMnyJihCB~lH nocTeneHHO yeeJIHllHBaeTCSJ H CTpeMHTCSI B npe.a.ene K ()ecKOHe'IHOCTH. llHCKpernafl cyMMa
8) ,n;u1
119
HHTErP AJI 4>YPbB
f
(t)
MOiKeT 6b1Tb san11caHa n n11.ne
!
co
!
00
/ (t) =
Cnei2r.vnt
An=
Cn'te-i2r.vt Jl'I,
'Jn= nvo,
-co
-oo
!l.n = 1,
s
Kor .na "t oqeHb neJIHKO, cyMMa nepexoJI.HT
HHTerpan
co t-
S
oo: /(i) =
C (v} ei2~• 1 dv,
(8. 21)
V=-CO
00
Jim (Cnt) = C (v) 1:-co
= S f (t) e-i2
r.• 1
t
dt = C* (- v). (8.22)
-00
B
HHTerpaJie (8.22) MH1erp11ponaH11e e .netic1n111eJibHOCTH npo· MCXO,llHT Ha HHTepnaJie 6, COOTBeTCTByIOULe~ O,llHHOQHOMy HM-
nynbcy,
raK
KaK /
(t) :;i:: 0
TOJibKO np11 -
6/2
< t K.ny
JIHHHSIMH JI.HCKpeTHoro cneKTpa p11c. 8.4. Pane1-1c1na (8.21) H (8.22) noKa3blB3lOT nOJIHYIO B3aHMHOCTb Mem.ny nepeMeHHblMH H v: ec1111 HMnyJibC / (t) MMeer cneKTP C (v), ro HMnyJibC C (t) HMeeT cneKTP ('J). Ecn11 HMnyJibC CHMMe1p11qeH, TO 11 cne1a1, yphe cnpaae,ZI.JIHBO paaeHcreo Tiapceea;n:t, CXO,llHOe c paaeHCTBOM (8.1 2), OTHOCHlllHMCH K PH.ll.Y 4>ypbe • .l\/\bl HMeeM:
j" If \2 dt =Sf*f dt =ff *dt SC (v) ei'211:vt dv. 3,ZI.ecb f (t) 6b1J10 aaMeHeHo cooTeeTcTeyromnM HHTerpaJIOM yphe Ha OCHOB3HHH (8.21 ). 11HTerpupysr no t H SaMeqas, llTO
l f * (t)
ei2.:vt
dv
= C* (v),
31
121
HHTF.rPAJI tl>YPbE
flpnuoyeoflbtfbtU uunylloc
8(v)
t(t} 1
_______..__..,__________ t
_§ 0
2
~
1 0 1 i - 281 28, T 1
1I
2
- e,
Cz11a}f(eHH01u 11uny11bC
e,
v 1
f(t)
-VH
-e,
tlv, 0
"l)H
_Jt12(j 0 ~ z
- 1
TpeyaOJlbHb!U 11MflYl7bC
f(t) 2
-(),
0
_fh_ i
lb_
e,
2
t
_L
t
o, -2l
1
1
2e1 tJ1
'aycfiafJ UMflYl1bC f t)
1
-20, Puc. 8.5. OpHMepbl cf>yHKunfi H Hx cneKrpos. t,
3TH rpaKJI.Y t u t dt, onpeJI.eJ1HercS1 KaK
l:i
+
p (t) dt =
~ If (t) I' dt,
(8.40)
rlle E ecr& no11nast 9HeprHH eOJIHhl, onpeJI.enHeMa51 (8.29), 113 KOToporo cJJe,llyer: co
sp(t)dt=l.
(8.40a)
-co
AHanor11qno, eepoHTHOCTb roro, qTo 4aCTOTa aaKJI10qena Mem.n:y v dv., pauna H v
+
(8.41) co
SP(v)dv=l.
(8.41 a)
-~
Tenep& cpe,nHee upeMSI noguneHHSI CHrnana ecT& 00
t0 =
Stp (t) dt,
(8.42)
-co
n cpe.llHHSI t1acr0Ta 00
v0 =
SvP(v) dv. -co
1)
CM., uaupuep, P. Ga b or, MIT Lectures, 1951.
(8.43)
oJ
COOTHOWBHHH HEOIIPHJIBJIHJIHOCTH
127
COOTHOl.lleHHHMH
t = f0
+t
v = v0
1,
+
"1·
CpeJIHHe KBaJI.parhI 9THx yKJIOHeHHtl paBHhl co
00
t: = .\ t:p (t) dt,
v! =
-co
Sv:P(v)dv.
(8.44)
-oo
Mb! onpeJI.e1u1eM cpeJIHHe OIUHOKH At no apeMeHH H Av no 1la· crore KaK
At=Vat!,
Av=-Vav!
(8.45)
H BblOHpaeM flOCTOSIHHyIO a T aK, lJTOOhl y .U.OBJieTBOpHJIOCb ycno· BHe (8.30a). HHaqe roaopsi, Mhl BblOHpaeM a TaK, llTOObl OhlJIO
(8.45a)
c
DOMOlllbIO (8.40), (8.41) H (8.44) nonyqHM BMeCTO (8.45a). co
00
a'
S t~f* (t)f (t) dt S v;c* (v) C (v) dv ~ E9.
i=·-CO
V=-CO
Bea norepu o6mHocru MO}l{HO noJJO>KHTh
ti= t
H '11
(8.46)
= V.
IlepBblM mar TO}l{,!leCTBa
B
t 0 = 0 H v0 = O; ror Jia
onpeJieJieHHH a COCTOHT B JIOKasareJibCTBe
co
u..
v9C* (v) C (v) dv = -
S 9=-oo
s
1 /* (t) d~ dtt dt, 4n: 2 t=-co
(8.47)
qro MO>KHO CJieJiaTb npH ycnoaHH, lJTO KaK / (t), raK H C (v) JI.HcpcpepeHu11pyeMhl H .llOCraToqHo ohlcrpo yohlna10T. HHrerpan 4>yp1>e (8.21) .naeT:
sc co
d2~ J dt
2 -
v==-w
(v) {- 4rc 2v2)
e'2nivt
dv,
l rn.
AHAJIH3 CHf HAJIOB
l28
8
H, T3KHM o6pa30M, 00
co
00
S /*~{.di=- 4it S 2
t=-oo
'V
~· f*e2r.M dt.
2
C (v) dv
t=-co
'11=-oo
HHTerpan no t ecTb He llTO HHOe, KaK C* (v) (cM. (8.22) ), u TO)KJI.eCTBO (8.4 7) 9THM JIOKa3aHO. IlpHMeHHH (8.47), MOiKHO nepenwcaTb (8.46) B BH]le
sr· !i. dt ~ I 00
(X)
- :;. s
t~* I dt
t=-00
t=-00
00
/*I dtr. (8.48)
s t-=-00
llnH JI.aJihHeDruero eocnonbayeMC51 HepaeeHcTBOM lilsapua
4
sF*Fdt s O*Qdt~I s (F*O+FO*)dtj
1 )
2
(8.49)
•
rJI.e
F=tf, BhtqncJIHH eTopoR MHTerpan no 1.1acTHM 11 yqHTbIBaH, qro MHTC51 K HY JTIO Ha oecKOHe 1u1ocTH, nony4aeM: 00
l
J
00
G*Odt=
-oo
s df* !!!_ dt-f* df dt dt dt
-oo
00
00
-
c1pe~
d 2f
Sf * -dt:!.d t =
-oo
- s!*di/
f
(X)
-
-
dt-:4
dt •
(8.50)
-co
VIHTerpwpyeM TaKJKe llaCTHM:
npaey10
qacTb
00
St(t*
HepaeeHCTea Weapua no
00
':ft +Id~ )dt=(tlfl >I 2
-co
-oo
-co
H HepaeeHCTBO Weapua npMHHMaeT Teneph BHJl (X)
co
-co
-oo
-4 1)
,lloKa3arenbCTBO KOHU.e STOA rnaBbl.
st~'ff* dt s/* ~'{. dt~E,,., HepaBeHCTBa illBapu.a .Z.HlHO
B
npttw1IO>Kem1tt
l:J
5]
129
COOTHOIJ!BHHE HEOnPEJIEJIEHHOCTH
CpaBHeHHe
c (8.48)
noKa3blBaer, qro Hy>KHO B3SITb
a2
= 161tg
HJIH
a= 41t,
onpeACJI5le~c
T. e. Mbl
HHTepecHo
cpaBHHTh
9TH
onpeJie.TieHHH
c
TeMH,
KOTOpb1e
(8.31) H (8.32). B npe.nll!eCTBYIOlllCM (8.25), (8.27) H (8.28) Mbl HMeJIH JJ,JIH~ 1/61. HOBble onpeJICJICHH5l JiaIO'f:
Bblpa)KeHbl COOTHOIJ!CHHSIMH paa.u.ene
JIJUI
TCJlhHOCTb
(8.25):
a.
npHMepoB H TIOJIOcy
E = 61,
V;
l::.t =
l::.v
61,
= oo,
l::.tl::.v
= oo;
(8.27):
(8.28): B
Bb1q11cneHHHX, npeJIOCTaBnHeMhlX
q0TaTeJ110, BCTpe4aJOTCSI
HH ..
TerpaJibl 00
sin x )2 ( S x . dx='TC,
-oo
-oo
-oo
Bo MHOf HX CTaThHX H pyKOBOJICTBaX no KBaHTOBOtt MexaHHKe npHHHMaeTCH
a= 2. 8. 1t=
Tor Jia
V
2t,2
= .y. rAt21t ,
EcnH TaKHe )Ke onpe.neneHHH npHHHTbl ll.JIH HollieHHe HeonpeJieJleHHOCTM npHHHMaeT BMA
~iqdJ]J;:;: It.
5
Jl,
J3pBJIJll03B
Av
8. 1v= -.r , , 21t
8. 1q
11
dJ]J,
TO coor-
[rn. 8
AHAJIH8 CHrHAJIOB
130
9r11 onp~neJietHHI npeJicTaBJI5llOTC51 HenpaKTHl.JHblMH B CBere npe,llLI1ecrBy10mero 06cynKHH5l rpatth AvAt JieJIHTCH Ha 21t. LlJIH cny1..1aH reJieBHJieHHH, ynoMSIHyToro B npell,hllIYIUeM pasneJie, 9TO 03Ha4aJIO 6hl, 4TO Jl,0CTaTOllHa noJioca OKOJIO 1 MKeHephl TeneBHJJ.eHH5l saMeHHJJH Ohl noJiocy 6 M21' rnecrhIO noJiocaMH no I Mzij. ecnH Ohl 9TO OhIJIO B03MO)KHO.
6. CTeneHH
CBOOOiI,bl
cooom.eHHH
PaccMorpHM reneph cne,n:y10W.Hti eonpoc: HeKoropa5l QJYHK~ UHH epeMeHH f (t) HMeer cneKrp, He COJiep>Kau1Hit 1..1acror Bhuue -v M, H cymecrayer Ha npoMe>KyrKe epeMeHH 't. CKOJihKO 11apaMerpoe (HJIH creneHeti ceo60JJ.hl) ueo6xOJJ.HMO JIJJR onpeneneHHst raKoti cpyHII raKoti cpyHKUHH HMeercH TOJihKO
(8.51)
N=2VM"t
>
(npelI.CTOJJaraH, 4TO 'J Af't 1) He3aBHCHMhIX napaMeTpOB, H OOCy· JIHM pa3JIH4Hbie B03MOlKHble BhI60phI 9THX napaMerpoe, a raKJKe HeJ{OTOpble o6mHe CBOtiCTBa paccMaTpHeaeMblX QJYHKUHii. 3aMeTHM npe2KJJ.e acero, llTO cpyHKUHH He onpeJieJieHa nonHOCThlO, eCJIH sa.uaHbl ee 3Ha4eHH5l TOJibKO Ha KOHellHOM HHTepBaJJe, HanpHMep 0 t 't. LlJIH KOppeKTHOti nocraHOBKH sana4H HyJKHO sanarb 3Ha 4eHHH QJYHKUHH lI.0 0 H llOCJie 't. IlpH 9TOM He Jl,QJI)KHQ JI06aBJl5ITbCff llTO-JlHOO K HHcpOpMaUHH, conepma .. w.eticH e coo6meHHH f (t). MOJ:KHO npHMeHHTh .uea pasnHllHhIX onpeJJ.eJieHH51. A. n e pH 0 Jl. H q e c Ka 51 cp y HK u H 51 I (t), oeCKOHe4HO noerop5110ma5I XOJJ. / Me>KJJ.y 0 H 't:
KJJ.eCTBe JlarpaH>Ka:
~
+
!
nM
n= I
cos ncp
= -2 1
+ nM
!
sin
•
e'"'P=
(nM + ~ h ,
2 sin (
n=-nM
r .ue 6epercsi q> = w0t. 8ra ¢YHKUHH paeHa n1u
i)
'
,
(8.61)
+ -21- rrpn t = 0
unn t = q't, rJJ.e aHaMenarenh paaeH HY n10. 0Ha KoJieonercSJ u oopaw.aetCH B HYJib B T04Kax
(nM + {-) w0t ='It, ~
t = N,
27t, ••• ,
21 m'C N , ••• , N ,
eCJIH m He SIBJISieTCSJ uenhlM.. KpaTHbl.M
mrr, .
m =/=. qN,
fJ...
:.
- ·-
6]
133
CTBnBHH CBOBOJlY COOBlUBHHSl
MY MoJKeM BBHTh JlJTH uaweD HMny11bCHOR
(8.64)
Senn
>t u 'tt
eeJlHKH, To v0 u
'1 1 npette6pemH~O MaJlbl
no cpas-
7)
135
MBTO.ll OTCqBTOB WBHHOHA
vMt
HeHHIO c BhlCrneft lJ3CTOTOf1
H Mhl HMeeM:
nonttoe qucno ro4eK orcqera cocraBnHer reneph:
(8.65) Vla lfHCna 9THX N 1 roqeK Mhl HMeeM N ro'leK, nonaJI.a10m11x B 11epeoHa'lanhHb1ti Httrepsan 0 t 't, H N 1 -N roqeK s HH· Tepsane 't OTC4eThl
vM,
2vM
•
(8. 70a)
137
81 OTKYJ13
I
cos 2rev td v =
g(t)=-2vM
1 -2vM --"'M
v=-"'M
f (t)
=!
ei21tv 1dv (8. 71)
+'JM
s
~:
m
ei2ttv (t-m6)
d~,
(8. 72)
-vM
06n.u1e np0HuHnb1 9TOfi npooneMhl OhlJIH oTKpblTbl HeaaeHCHMO HeCKOJibKHMH yqeHblMH. 0epBOH3lJ3JlhHaSI HJI.eSI npHH3JlJie)KHT, DO .. 2
BH.llHMOMY' y HTTeKepy ( 1915 r.) I, ). llaJibHefirnHe Il0Jlp06HOCTH H npHJIO)KeHHSI Jl3Hbl B paa.nenax 6 H 7 r Jl3Bhl 17.
8. Hu(f>opMaUHOHHhre gqeifKn faiioopa Ilsa paCCMOTpeHHhlX npeJICTaBJieHHSI SIBJISHOTCSI qaCTHblMH CJIY .. 3 1.1asiMH oonee o6w.ero Mero.na, paccMorpeHHoro f9fi6opoM ). l
a
3/1
I
0
I
'(
R11eltxo '1. v ·J t • 1
-- t
V1 !/'It
-- t
0
Puc. 8. 7. PaaJJH'IHbltl: &bloop HlJeeK eJJ.HHH'IHOil DJIOlll8JlH ua nnocKOCTH v, t:
-v,t
Ha nnocKOCTH o6nacrh, aaHHMaeMaSI ¢yHKUHefi, eCTh npsiMoyronbHHK 'V M't (pHC. 8. 7). 3ror npSIMOyrOJibHHK MOiKeT 6b1Tb paa]leJieH Ha 9JieMeHrapHhle HqefiKH e.nHHH4HOfi nnow.a.nH
llvllt=l, H B Ka)K.nOfi siqefiKe
¢yHKUHSI HMeer .nae COCT3BJISllOW,He -
(8.73) CHl\f-
1) E. T. W h it taker, Proc. ~oy. Soc.t Edinburgh 35, 181 (1915). 2
B 01t1er ne:eoil H coepeMenHoil cJ>opMe pa3nolKee:e:e (8. 70) )lano B. A. KoreJJbHHKOBblM e 1933 r. (flpu.M. nepeB.) 3 ) D . 0 ab or, Theory of Communication, J. Inst. EL Eng. 93, Ill, 429 (1946); 94, Ill, 369 (1947): CM. 1aKlKe Phil. Mag. (7] 41, 1161 (1950): La Cybernetique, p. 115, Editions Revue d'Optique, Parist 1951. )
138
AHAJUl3 CHrHAJIOB
[rn ..
8
MeTpH4HYIO H aHTRC11MMetpHlfHYIO, - .zise 9JieMeHrapHbie cpyHKUHH, AMnJIB~ HCnOJib3YIOlllHe m1-1pHHY nOJIOCht d'J H .zinHTeJihHOCTb Ty .llhl ooeux cpyHKUHtl .llOJl'.IKHbl Obi Tb 3aD.aHhl H nOJIHOCTblO onpe .. .ziensnor Harny cpyHKUHIO (t). 411cno siqeeK paauo
at.
f
n=
YM't•
l.J'HCJIO COCT3BJISllOillHX
N= 2n= 2vM't,
(8.7 4)
KaK a (8.51 ). opMa HlfeeK He Hrpaer ponH; HMeer aHa4eHBe TOJihKO HX IlJIOllla.llh. P51D. KeHue (8.80) COJiepiKHT ronhKO nM 1 cTeneHeR ceooonb1, TaK KaK MbI n0Tep>1nu nM cpaa. Or paano>KeHHfi Ke1 OblTb eoccTaHoeneHa no oTcqeraM, KaK
+
B
(8.60): nM
R(t 0)=
,!
Rm[g(fo-m6)+g(t0 +mfl)J+R0g(t0),
(8.86)
m=l
r Jle tj>yHKUHSI g· onpeJieJieHa paeeHCTBOM (8.62) . N. Wiener, Acta Math. 55, 117 (1930); A. K hi n tchi ne, Math. Ann. 109, 604 (1934); Y. W. Lee, T. P. Che at ham, J. B. Wiesner, MIT Report N2 141, 1949. 1)
I 0)
141
nHHBRHYB nPB06PA30BAHHH A ~HJihTP~
+
9Ta npouell.ypa HCnOJib3YeT TOJibKO nM 1 CTeneHeit CBO· 00.ll.bl HCXOll.HOA QJYHKLUUI H3 o6mero lJHCJia 2nM 1. 0Ha c. ycnexoM np11MeHs:1eTCH ao scex sall.a11ax, e K0Topb1x cpaahl rpn HeH3BeCTHhl JIHOO no npHllHHe HX CJiyqattHOCTH (npo6JieMbl rnyMa), JIH60 BCJieJlCTBHe roro, tfTO H3MepHTfJihHblt1 npouecc He B coCT051HHH 06HapyJKHTb QJ330Bhle yrJIId (HanpHMep, HCCJiell.OBaHHe crpoeHwsi KpHCTannoa peHTreHOBhlMH nyttaMH). BMecTo KOMnneKcHoro pHll.8 ct>yphe (8. 76) ll.JIH (t) Mbl MOJKeM eocnoJih30eaThCH pHJlOM ct>yphe s aemecTBeHttott cpopMe (8.54):
+
f
nM
f
(t) =
-~ Ao+
,!
An cos (nw 0t
+ cpn), (8.87)
n=I
H cor JI8CHO
(8.80) nM
R (t0) =
! Ai+ ~ !
A: cos (nw0t 0).
(8.88)
n=J
Benh1tt (TenJiosott) lllyM, HanpHMep, ll.aeT nocTOSIHHOe SHa11eHHe A~, He 38BHCHIIJ,ee OT n.
to.
JlHHeHHble npeoopaaoBaHHB H l)lHJILTphl
MOIIJ,Hhlf-1 MeTOll. HCCJiell.OBaHHSI cpyHKilHtt OCHOBaH Ha JIHHettHiilX npeoopaaosattHHx 1). PaccMorpuM cpyHKUHEO H ee HHTerpan
H. Lab {oust e, l. Lab roust e, Comptes Rend. 184, 259 (1927); Ann. Inst. Phys. Globe Univ. Paris 7, 190 (1929); 9, 99 (1931); 11, 93 (1933): Terrestrial Magnetism and Atrn. Elec. 41, 15, 105 (1936); H. Dev Tables nurneriquest Evreux, Paris, 1930; M. Levy, Comptes Rend. 198, 2222 (1934); 199, 1031 (1934); 200, 646 (1935); Ph. D. thesis, Paris, 1951. H. Labrouste 3aHHMa.1Jcs:1 cneu;HanhHO cnyt.1aeM tJHCneHHoro B.blt.1HcneHHH no JJ.HCKpetHblM suatJeHHHM ynKUHH. OH rronh3osancH o6lJ.lHM npeoopasoBauneM BHJJ.a 1)
e,
F (t)
=!
"
Cfnf (t
+ nt),
r)le n - u;enoe. M. Levy 0606lJ.lH.rr peJynbTarbl JJ.JIR nurerpanbHoro npe06pa3osauH11 BHJJ.a (8.91).
142
Ll'Jl. ~
AHAJIM3 CHrHAJIOH
f (t)
=Sc
(v) e2"ivt d"
(8.89)
i ('1) e2r.M dv,
(8.90)
B ¢yHKUHIO npeoopa30BaHHH
~ (0) = ll.OCTaTOttHO
ObICTPO onpe~ensercs KaK
npeo6pa30BaHH8~
YOb1BaIOmy10.
F(t)= u
s
st(t+8)~(8)d6
cpyHKUHSJ
(8.91)
Mo:>KeT OhlTh npe.u.cTaeneHa MHTerpanOM yphe
SC (v) e dt = Sf (t') e-
12"' 1 dv,
F (t) = r~e
C(v) =
S
Fe-12-ir.•t
12 "' 1'
(8.92)
S~ (0) e+12r.,e dB. t' = t + B.)
dt'
(3.nech HcnonhBOBaHO (8.91) H nonoiKeHO paBHhl COOTBeTCTBeHHO C (v) H (v), TaK 4.TO
r*
YIHTerpanhl
C(v) = c (v) r* ('1).
(8.93)
TaKHM oopaBOM, npuMeHeHMe cpyHKUHH npeoopaBOB3HHSI r.p 9KBHBaJJeHTHO cf>HJlbTpauuu npu flOMOIIJ.H cf>HJibTpa c nepe.u.aT04.· Hoit cpyHKuueii T('v). JlaopycT u Jlesu pacCMOTpenu pSI~ cpyHK.. UHii npeoopaBOB3HHSI, DOBBOJl5110lll,HX Bbl.lleJIHTh CHHyCOH,ll3JlbHble COCTaBJl5IIOlllHe HB CJIOiKHhIX KPHBhIX, nonyqeHHhIX nyTeM 9KCnepHMeHTa. Pa3JJH4.Hble HMnynhChI (8.25)-(8.28), HanpuMep, ~aIOT nepe~aT04.Hble cpyHKUHH, COOTBeTCTByIOmHe HX cneKTpaM, H MOryT npHMeHHThCH ,llJISI npeo6pa3osaHH5I. yHKUH51 (8.91) eCTh npocro B3aHMHa51 KOppenHUH51 MeiK,ny cpyHKUH51MH 11 r.p. Kor.na r.p COBrra~aeT c /, Mbl nonyqaeM aBTOKOppeJJ51UHIO cpyHKUHH /, H paBeHCTBa (8.91) H (8.93) CXO~Hhl c (8.77) H (8.79). Yro6hI noKasaTh CB513h MeiK.llY pe3y nhraTaMH pasnH4HhlX pa3.u.enos 9TOii r naehl, paCCMOTPHM npHMep. Ilpe.u.noJJO>KHM, 4.TO Mbl paCCM3TpHBaeM 9KCnepHMeHT3JlbHYIO KpHByIO (t), OTHOCHTeJlbHO KOTOpoii HBBeCTHO, 4.TO ll3CT0Thl Ilhlllle HeKOTOPO" VM sanucaHbl ne OlleHh Ha,lle>KHO. y crpaHHM, npem.u.e Bcero, 9TH ll3CTOTbl, qTO MO>KHO c~enaTh npu noMow.u npeoopasouaHHSI (8.91} c cpyHK·
f
f
nHHE~H~H UPEOSPASOBAHHH n •nn&TPbI
10]
UHe~ npeoopaaoeaHHH
F (t) yHKUIHl r.p
(6)
(8.26);
Ayncuo eblqncnnTh
sin (2rcv MO) B) 2rcvMa dB.
= 2vM S/(t +
f l
1
npH
0
npu
Iv I~ "M' Iv i > "M·
Ilpeo6pa30BaHHe npe.ncraBnHet H.neanhHhiti 11
(8.94)
peBKO orpaHHqeHa Ha qacTOTe VM; ee cneKtp
1 (v) = CTOT
143
UOJJ,aBnHeT
Bee
BhlCOKHe
Hnhtp
qacTOThI.
HHJKHHx qa-
OHO
er Jia)l{HBaet
9KCnepHMeHranhHYIO KpHey10 H yctpaH5IeT OCTpbie yr nhI H pbIBhI, Kotopb1e
F (t)
MoryT
-6hITh
coMHHtenhHhtMH.
HoBasi
pa3-
cpyHKUHH
11MeeT otqeTnuey10 rpau11uy npu VM.
EcnH Haonio.neHHe Benach aa
apeMH
cpyHKUHH
tt,
co.nepncH'f
TOnbKO
N= 2\IM
't
cteneHe~ caoooJJ.1>1 (cM. (8.51 )). Mhl paccMarpusaeM ee KaK cpyHKUHIO OJI,HHOqHoro coo6meHHH (pa3.nen 6, B) H npHMeHHeM MeTOJJ. orcqetoB WeHHona (paa.nen 7) c HHtepBaJioM otctt:e·roa
(8.68): l Bum=-2-· VM
OnepauHH
OTClfHT&IBaHHff
cne.noaaTeJibHOCThIO
FM= F (m9um).
3aMeH5Iet
HMny Jib COB
c
uenpepbIBHYJO OTC qHTaHHbIMH
KpMBYIO
no-
aMnJIHTY JJ.aMH
nocJieJI,OBateJibHOCTb HMnyJihCOB Hl\leet renepb yKa3aHO
B
pa3.n;eJie 1
(8.9), (8.11) H p11c. 8.1 ). CneKTp COCT011T cneKtpa (Iv I~ "M), nostopHeMoro nepHo.n;H4ecKH
HB
HCXOJJ,HOro
oeCKOHellHO
npOT.sJJKeHHblti
cneKTp, KaK
(CM.
qacTOT c nepHO.ll;OM 2YM. Otcllethl Moryt nepe.naeatbCH no cHcteMe MeTOJJ.OB COBpeMeHHOti HMilYJihCHOti texHHKH.
B)I.OJih wKaJihI
CBSIBH
OJI,HHM 113
3rH 0TC4eTw BOC-
np0113BO..ll.HTCSI Ha np11eMHO~ CTaHUHH, H Ba,aaqa COCT011T
F (t).
CTaHOBJieHHH HenpephIBHOti KpHsott 9JieKtpHqeCKHM nyTeM JJ,eJI5IJOIUero OJI.HY
113
npH
DOMOIUH
OJJ.HH3KOBhlX
8to
MO}f{HO
IlOJIOCOBOro
tiaCTOTHhIX
B
c.nenath
cpHnbtpa, BbI-
DOJIOC
p11c.
To )Ke caMoe MO)KHO c.nenaTh u nyreM BhI4HCJieHHSI cyMMhI
F(t)- ~ F
-~ m
sin m
'lt
(2vM t-m)
rc(2v~-wt-m)'
BOC-
8.1.
(8. 70): 8 94 (.a)
144
frn. 8
AHAJIH3 cnrHAJIOB
qTO CHOBa COOTBeTCTeyeT MaTeMaTHqecKOMY OTIHCaHHIO neUCTBH~ HJJ.eanbHOro cpHJibTpa HHiKHHX llaCTOT. .[lnSl BhlllHCneHHSl cpyHKUHH e MOMeHT t HaM ey:>KHbl e npneuHne ece OTClleThl JJ.O H nocne t. EcnH Mbl XOTHM BOCCTaHOBHTb F (t) c TO'IHOCTbEO .no 10-1, Mbl .noniKHhI pacnonaraTh no MeHhrnett Mepe 1O' qneHaMH JJ.O H nocne MOMeHTa t, TaK KaK qneHbl pa3nOJKeHHH y6b1BaEOT OlleHh Me.nneHHO, ecero nnmb KaK m- 1 unu (t' - t)- 1 • OJJ.HaKo eb1llHcnee11e BblllienpHBeJJ.eHHOlt cyMMbl SIB.TIHeTCH, nO~BHJlHMOMY' HaunyqrnHM nyTeM K BOCcTaHOBneHHIO KpHBOit, ecnH TpeoyeTCSI BblCOKaH TOqHOCTh.
1 1.
AnaJIH3 opMynhl 51BJH110TC51 npSIMbIM OOo6meHHeM u (8.3). CooTseTCTB11e oe5oaHa4eHHi1 TaKoso:
't:
OCHOBl-la5I 4aCTOTa Vo:
d 1, d 2, d 3, b1, b-}.1 b3,
HOMep rapMOHHKH n:
h1' hi, ha.
nepnoJI.
(8.1 ), (8.2)
(rn. 8
AHAJlH3 CJtirHAJIOB
146
B TpexMepHoM cneKTpe Moryr ecrperuthCH pasnaqHoro po4a orpaHHqeHHH.
A. Tpu omoeJtbHblX npeoe1ta 8 mpex Hanpa8JteHllJlX
I h1b1 I ~Pt, Ih.,,b.,, I ~ P2' I haba I ~ p3,
{8.99)
OTKyna (CM. PHC. 8.11)
l h1 I~~;= P1d1, I h9. !~ P2d2, Iha ~ pada. J
Be.rr11QHHb1 p1 p.,, p3 npe.n:craBJIHIOT HaHBhICll.IHe qacTO'fbI B nanpaB· neHHSIX 1, 2, 3. 3a.llaqa npe.n.CTaBJISieT COOOii nonpocTy HaJIO>Ke· nHe rpex oT,n:enbH~x o.n:HoMepHMx aa,n:aq THna A (CM. paa.n:en 6 !/J
Y2
PHc. 8.11. 06parHafI pernenca onpe· ]leJlfleTCJI BeKTOpaMH bi, bi, b,,. To'IKH pemeTKH JJe>KaT BHyTpH noKa3aHHoro Ha
p3 -
pucynKe nap&JJJJeJJemmcJl.a, ecJm p1. p2, HaHBbICLUHe qaCTOTbl B Tpex M3MepeHlH1X.
9TOA r.rraew). aHa qeHuii:
B
PHc. 8.12.
06patHaH perneTKa onpeJle.TJeua BeKTopaMH bi, b:z, bi. p He aaBHCHT oT TO'llKH petuCTKH BHC pa.1rnyca p HCKJHOlfCHbl.
Hu160.11&ll1aH qacToTa
HanpaBJJCHHSI,
cepb?
I{
sroM rrpHMepe HMeeM TaKoe cootaeTCTBHe ooo-
HaHBbICIIIas. qacroTa
'i M • • •
P1P2Pa·
B. PaccMorpHM eiJUH.blU npeiJeA ttacmombl p npH yc.rrOBHH (8.100) (pHC. 8.12). 9TO ycnOBHe COOTBercreyer KpaTttaitrneii .ll.JIHHe BOJIHbI 1/ p. Oocy ,n:HM no,npo6Hee ero cp11aHqecKwtt CMhICJI. Kan~ .. n.aH cocrae.rrsuomaH h1 h.,, h3 e paano:>KeHHH (8.97) Bb1paiKaeT
11 J
AHAJIH3
~YPbB
e
cuHycoK.ll.8JibHOe pacnpe.lleneHue Jie)Kamne B IlJIOCKOCTH
h1b1X1
147
H METOA OTC4ETOB npocTpaHcTse.
Bee
T04KK,
+ h2b2X2 + hab3X3 = 0,
HMelOT TY )Ke cpasy, ttTO H B Ha4aJie. Llpyrne napaJIJieJ1bHh1e rIJIOCKOCTH c OAMH3KOBOii cpasoii HaXO.llSITCSI l:f 3 COOTHOWeHKSI
h 1b1x 1
+ h9.b9.x'J + h b x 3 3
r.rr.e ni - uenoe. llJinHa BOJIHbI coce.rr.HHMH flJIOCKOCTSl~1H dm
3
= m,
A
=
ecrh paccroHHHe Mem.ny JJ.BYMH -+- 1, H JiefKO HatiTH:
(8. IOOa) r JI.e p onpeJI.eJieHo (8.100). DOJlhlllMe SHal:JeHHSI h COOTBeTCTByIOT KOpOTKHM BOJIHaM. BeKTOp (h 1b 1, h'1.b 2, h 3b8) opToroHaJieH nnoCKOCT5IM BOJIH, H ero JI.JIMHa eCTb 1 l,
ecnu p1p2pa
p'Jd'J>t,
MHoro OOJibIIIe,
Pada>l.
Il poueJI.ypa OTC4HThlB8HlUI, OTIHCaHHaSI B pa3JJ.eJie 6 (CM. (8.59)-(8.64)) 1-1enocpe.llCTBeHHO npnMeHKMa npH np5IMOyroJihHOfl perneTKe c N T04KaMH OTc4eta BHYTPK SJ4etiKH d 1d 2d 3: Xis=
m1e., (8 .. 103)
148
(r11 . 8
AHAJIH3 CHf HAJIOB
3.nec& UMeeTCSI COOTBeTCTBHe: HHTepeaJI OTC'IeTOB l1cXO,llH8SI cpyHKUHSI nerKO npH IlOMOIUH TpOi1HOit cyMMbI
F=
!
6, •.• ,
e., ;2,
;3•
BOCCTaHaBJIHBaeTC51
no
OTCqeraM
Fm 1m2 m3 gt (Xi - m1e1) g'I. (x2 - m2E2) X (8.104)
rJJ.e
Fm 1m2m3 = F (m1~1t m2e2, msea). yHKUHSI
g
Oh1na onpe11.e.11eHa paBeHcTBOM
g;
( X; )
=
sin (N;1tb;xi) N· sin (1tb·X·)'
'
(8.62):
i= 1, 2, 3,
(8.105)
' '
ecnH npHHHTh BO BHHMaHHe COOTBeTCTBHe OOOBHaqeHHtt. 3aAaqa, TaKHM o6pa30M, UOJIHOCTblO peweHa, H MeTOJJ. OTC'IeTOB npH .. MeHSieTCSI OeB BCSIKHX BaTpy.nHeHHit. BTopaH saAaqa, B, c eAHHCTBeHHbIM npe.nenoM p (CM. ( 8.100)) 51BJISieTCSI oonee CJIOJKHOit. Mbl nerKO Hai1.neM KOHelIHOe qHCJIO creneHeit CBOOOAbI, HO MeTO,U OTC'IeTOB MOJKeT 6hITb JIHWb Ha· MeqeH H OCTaeTC.51 npe.nMeTOM o6cy)KJJ.eHHSI. CocTaBJIH1011u1e KHT TOJihKO
npe.u:cTaBJJeHHH
Heno..
cpyHKUHH
He3anpH
nOMOWH OTC4eTOB HeOOXO,lLHMO HMeTh B CHJIY CHMMeTpHH TOJlhKO
I
2
N
oTctJeToB Ha npoTH>KeHHH nonoBHHhI 9JieMeHTapHoti KpHCTan-
JIHtJecKoti HtJetiKH. 1
Jlpyroti MOWHhlti MeTO.Q HCCJJe.QOB3HH51 npe.QJJO>KeH nenHHCKHM ) flO.lL H33BaHHeM
KOHBO.ll10f;'1llU.
npeoopa30B3HHHM, .QJISI
Tpex
npaKTH4eCKH
OOCyJKJLeHHhlM
H3MepeHHti.
8TOT
B
OH
pa3.QeJie
MeTOJI.
COOTBeTCTByeT
10,
OK333JIC51
OOOOWeHHhIM
HCKJII04HTeJihHO
none3HhIM npu 11ccne.u:oaaHHH cnoJKHhIX CTPYKTyp. MhI peKOMeH.u:osan11 B KOHUe
(8.28)
pa3.u:ena
10
npHMeHeHHe
B Ka4eCTBe H.U:ep JIHHetiHhIX
H33hIBaeT cpyHKURIO
HMTIYJihCOB
(8.25)-
npeoopa30B3HHti. nenHHCKHti
(8.26) aapo.M Jl,tipllX.lle
H 00p3THYIO cpyHK·
(8.27) - aapo.M o.
(8A.2)
BemecroeHHoro KopHH He JlOJIJKHO 6bITb; no9TOMY B 2 -4AC~O,
HJIH
(8A.3) a 9ro u ecrb Hepaoeucroo Woapua.
fJlABA 9
OCHOBbl
TEPMO~HHAMHKH
t. B
BseJleHHe
npeJlhIJI.Yll~MX
r naBax o6cym.u.ancff Bonpoc o TOM, KaK MaTeMaTH4eCKM onpe)leJIHTb MH¢opMaUHIO " K3K H3,\1epHTb ee KOJIHlleCTBeHHO. OrpaHnlleHHOCTh TaKoro onpeJI,eneHMSI 6hlna noJl· qepKHyTa B nepBo" r naBe; nocne.u.yrowne r naBhI 6hlJIH nocBHWeHbl paaBHTHIO " np11MeHeHHHM o6wnx onpeJleJieHM". oblna o6napymena TecHaSI CBSI3h Me>KJlY TeopHe" HHcpopMaunn H TepM0JlHH3MMKOtt; HHcpOpM3UMH npeJI,CT3BJI51eTCff CBSI3aHHO" c 9HTponnett. 06cymJleHne 9Toro HOBoro acneKTa Teopn.w MhI HatIHeM c KpaTKOfO o6sopa OCHOB TepMO.llHHal\tHKH.
2.
~Ba
eaqaJia
TepMOJlHHaMHKH;
9HTpODHSI
H eer9HTponHJI
A.
CoxpaHeHne 9Heprnn. B. CTpMHUHil Kapuo. no KeJibBHHY 9TOT npHHuMn 03Ha4aer iJe2paiJaqrt10 81tep2Ull.
3HeprHff BhICOKoro KalleCTBa (high grade)- 9TO MeXaHHIJe"' CKaH H 9JieKTpMqecKaSI 9HeprHff. 8HeprHH cpeJlHero KaqecTBa - 9TO XMl\IHlJ:ecKaSI 9HeprHSI. 3HeprMH HM3Koro KaqecTBa - 9To Tenno . 06IIlee KOJIHqecTBO 9HeprHH OCTaeTCSI HeM3MeHHh1M B 33MKHY· Tott H30JIHpOB3HHO" CHCTeMe.. CTpeo6pa30B3HMH HJIH XHl\fMlJ:eCKMe peaKUMM eHyTpM cncTeMhl He HSMeHffIOT KalleCTBa aHeprun npu o6paTMMhlX npeoopa3oBannsix. Heo6paTMMhle npeo6paaoBaH11H noHM· iKaIOT KaqecTBO. KaqecTBO 9HeprHH 1\tomeT 6hITb Toqno onpe.lleneuo qepea ompuqamellbHYIO BHmponu10 ( - S). CornacHo eTopoMy Ha4any 9HTponHst Beer .lla B03pacTaeT, MJIH no KpattHe" Mepe ocraercsi HensMennott, TaK llIO orp11uaTeJibHaSI aHTponHff ocer Aa y6h1Bae-r.
155
2] OnpeneneHne 9HTponnH
Ecnn CHCTeMa nony1:1aer KOJIH4eCTBO Tenna ll.q, TO 3TO COOT• seTCTByeT ysenH4eHHIO 3HTpOnllH CllCTeM~ Ha :lq
AS= T'
(9.1)
l1q= TAS,
T - aocOJIIOTHaH . TeMnepaTypa (no KeJJbBHHy).
B
rpanycax
UenhCHSI
T°K=t 0 c+213,15. HecKOJJbKO THilHl[HblX cny1:1aee nocnyiKaT npnMepaMH. PacCMOTpHM nse conpwKacaIOru.Hecst CHCTeMhI A H B, Morymne ooMe· HHBaThCH paooToti " TennoM. By neM C4HTaTb, l[TO cncTeMa AB naonHposaHa OT csoero oKpyiKeHnH. HeKotopoe npeoopasoeatt11e npHBOJI.HT K cneJI.ylOW.llM ycJJOBHHM: CosepwaeMaSI pa6oTa
(work output)
IloJiy"laeMoe TenJio (heat input)
TipHpaw;emi:e 9HTponHH (entropy input)
A
B
nonHaH 9HeprHH CH CTeMhl AB OCTaeTC5l Hell3MeHHOA. IlpuMep~
ueoopaTHM~X
npeoopa30B3HHA
llomoK menJta OT ropgqero Tena K xononHoMy: Paoora .He coneprnaeTcH: WA= O; W 8 = 0. Ilepe.u.a.HHoe Tenno: qA=-qB, qB>o
AsA+ASB=qB( 06111aB 3HTp0llHB B03pOCJia.
iB - iJ>o.
(9.3)
156
OCHOBM IBPMOAHHAMHKA
[rn. 9
Tpe1tue, Bfl31'oe iJe.MnifJupooa1tue: TA= T n = T- o.n.Hopon·
Hasi TeMnepaTypa. PaooTa cosepwaercsi TenoM A, Teano BMJI.ensieTcH e B:
wA>o, qA=o, WB=O, qB>o, WA-qB=O, qB
ASA+ ASB= T
WA
=T
> o.
(9.4)
0 omaH 9HTpon1u1 sospocna. noJIHaR 9Hepriu1, co11epJKamaS1cS1 e cncreMe, Momer o&ITb onpe11eneHa, ecn11 MM sHaeM noc1poen11e c11creMM na ee cocTaB· HbIX qacyelt (9HeprHSI KOTOpMX npeADOJiaraeTCSI H3BeCTHOt;l) H ecnu Mbl ytITeM coaepwenny10 npn nocTpoeHHH pa6oTy n BMJI.ennB· weecS1 Tenno. IlonHaSI 9Heprnsi U ecrh cpynKunH scex qn1a11tJe· CKHX H XHMHlleCKHX napaMeTpOB, onpe.n.enSIIOIIlHX CHCTeMy. Ecnn c11creMa MOJKeT 6hITb nocTpoeHa nyTeM nocJieJI.OBaTenhHhIX otJpamuMbl.X npeo6pasooa1tuu, TO no1t1taR aHmponuR CHCTeMY onpeAenHeTCSI KaK
s
(9.5) T. e. KaK cyMMa 9HTpOOHfl, CBSl3aHHblX C K81K.ll.hlM oopaTHMMN waroM npeoopaaosaHnsi. HezaHmponllR 1) (N S) npeJI.cTasnHeT 1'attecmoo aneprHH, u .ll.OJI>KHa seer .n.a yoMBaTb. B 3TOM COCTOHT CMblcn KenbBHHOBa npHHUHila OezpaOall;UU 81tep2Ull, H Hey.ll.lfBHTeJibHO, lJTO eaJKHOCTb Her9H'rpon11H oMna npnanaHa onH3KHM JI.pyroM KenhBHHa, KOTOpMt;l nucan 2): «Bec&Ma JKenaTenbHO HMeTb cnoso, Bb1pa)Ka10wee cnocoo· 1tocmb Tenna, aanacenHoro e HeKoTopoM CKJia.n.e, nponasecTn paooTy; repMHH .n.nH B03MOJKHOCTH, yTepsi KOTOpolt H83MBaeTCH pacceJ11tlleM. K co)KaneHnlO, npesocxo.n.Hoe cnoeo aHmpon1in,, KOTopoe KnayaHyc aaen s CBSl3H c 9THM, OhIJIO HM npHMeneHo B 0 T p H u a T e JI b H 0 M CMhICJie no OTHOWeHHIO K IlOHBTHIO, KOTO• poe uaM ecTecTeenuo xoqercsi BMpasntb. Ho MM JIHWh cMy .. THJIH OM y11awerocS1, ecnH 0&1 nonhlrannch BhIJI.yMaTh nosbltt TepMHH AJISI 9Tott uenn. BMecTe c TeM HaJI.OOHOCTh B TaKoro poAa Tep·
=-
1)
CoKpameuHe
JIJU:I
negative entropy- orpnuare.11huott serponHH.
(npu.u. nepeB.) s) P. G. Ta i t, Sketch of Thermodynamics. p. 100. Edmonston
Douglas, Edinburgh, 1868.
3] HBBOSMOiKHOCTL BBqHoro JI.BIDKBHMst; TBnJJOBbIE MAWHHbt 157 MHHe 6y D.eT OlleBHll.H8 M3 npeKpaCHhIX npHMepoB, cne.nyromHx HMiKe». MaonMponaHHaSI cMcTeMa oona.naeT HezaHmponueu, ecnH oHa oouapymMsaeT noaMOiKHOCTh coneprneHMSI MexaHttY:ecKott MJJH 3JJeKTpMY:eCKOtt pa60TbJ: eCJJM CMCTeMa He MMeeT OJI.HOPOJI.HOtt TeMneparyp&I, a COCTOHT H3 pa3JJH4:HbIX llacrett c pa3JJMllHblMM TeMneparypaMM, OHa co.nepiKHT HeKOTOpoe KOJJHqecTBO HerSHTpO· """· HerauTponMst MoiKeT O&ITh Hcnonh3oeaua lf.JISI nonyt1eHMH HeKoTopo" MexaHHY:eCKOtt paOOTbl, conepwaeMOtt CMCTeMott, MJJM oHa MOiKeT OhITh npocro pacceSIHa M norepstHa ncne.ncrnHe renno ... npOBO.llHOCTM. Pa3HOCTh JI.anneHM" n pa3JJHY:H&IX qacrsix cHcreMbI HBJJsiercst .npyrMM cnyt1aeM HerauTponMH. Pa3HOCTb sneKTpHt1ecKHX noTeHuHanon npe.ncTannsieT eme Oll.HH npHMep. Cocy.n co ciKaThIM B03JI.yXOM B KOMHaTe npM aTMOCcpepHOM JI.aBJJeHHH, OTK3ll8HHblA cocy JI. n TaKott iKe KOMHaTe, sapstiKeHHast 6aTapest, nioooe ycTpottCTBo, KOTopoe MOiKeT rtpOH3BOJI.MTb 9HeprMIO BhICOKOro KalleCTBa (MexaHHY:ecKyro paoory) MJJM JI.erpa.nMponaTh B pe3ynhTaTe KaKoronttoo HeoopaTMMOfO npouecca(rennonpOBO.llHOCTb,9JleKTpM4:eCKOe conpOTRBJleHMe, Tpeuue, BSISKOCTL ), ecTb HCTOllHMK HersHTponHH. BaiKHOCTh paccMorpeHHSI OTpHuaTeJJhHott surponnH OhIJJa nO.llllepKHyTa Wpe.nttHrepoM e ero HHTepecHettmett KHMre KEHHE 11 6POYHOBO JlBH>KEHHE
1. Tennosoe neu>KeHue HanOMHHM O.ll,HO ea Ba)KHhIX aaKJHOlleHHtt npe.ll.hl.llymeti r naBbl, a HMeHHO, 11ro npH
HH3KOtl
-
Ekin=
Q3CTOTe,
1 2 H
kT
JlJHI
KaiK,llOtl
npO,llOJJ)KHM
creneHH
QOCYiK.ll.eHHe
ceo60.llbl
HeKOTOpbIX
npaKTHlleCKHX npHJIOiKeHHtl ..
Bpoyttoeo llBHiKeHHe
Ha6JJ10Aa~
MeJIKHe
llaCTHUbI,
norpy»KHbJ .nea paeHoeeposiTHblX co6b1THH - n060 qacrwua 0cnycKae1 csi, JIH6o Her (p11c. I 0.5). TaKHM o6pa3oM, Ha nporH>KeHHH sroro HHT€pBana H3MeHeHHe 9MHCCHH eCTb
A.t
= 6,
(
o AP= {
1
1--
1==++1+ }·
orKyna cne.llyer, 'ITO
A.v=
--~
!l
1•
·2
(10.13)
178
T.BllJIOBOH JlBH'iKBHHB H BPOYHOBO ABH>KBHMH
TaK, ecnH BSHTh
I l I= -21 ,
[rn. 10
TO Mht CBOJUIM 3ana'ly K nocTOSIH-
HOMY HapacTaHHto Nt, Ha Koropoe HanoJKeHo CJJyqattHoe 6ny>KJiaHHe. CpeJIHHtt KBa11par cpJiyKryauHtt no-npemHeMy Bbtpa· maercsi KaK ·-·1
-n/2-4- n,
{10.14)
2 Vn -
rne n BhlpamaeT noJIHoe BpeMSI t = nO. Ha nporsiJKeHHH eroro npoMe)KYTKa BpeMeHH t cpeJIHSISI 9MHCCHS1 paBHa
-
P
= Nt =
l
n6 26
= 2l
n,
oriJiyKryauHH e paccMa-rpHeaeMoM cny11ae paseH nOJIOBHHe cpenttero 4HCJia 9.MHTHponaHHblX 4aCTHl1. BoJiee ea>KttaH 11 6oJiee 6JIH3KaH K neflCTBHTeJibHOCTH cpH3H· 11ecKaS1 sa.na11a nonyqaeTCH npw HHblx npennonoJKeHHSIX, cooTBeTCTsyiomwx no;uioc mb10 cAy'launoa a.Muccu11. MbI BbI6upaeM ottettb MaJihltt epeMeHHOtt wurepeaJI 6 H npennoJiaraeM, ttTo ua npoTSiiKeHHH 9TOf O HHTepeaJia npHpameHHe 9MHCCHH ll.P ecrb
ll.P= {
0 c sepoS1THOCTb10 1 - NB, 1 c sepostTHOCThIO NfJ
< 1.
a
HHTepBaJI H3CTOJibKO Mall, ltTO 9MHCCHH JlByx, Tpex H 6onee ttaCTHU cqu raeTCSI HeBo3MOiKHOtt. CpenHHSI CKopocTb 9MMCCHH eCTb N. Y cnoBHe MoJKeT 6bITh aanttcaHo a c11e11yIOmeM BHJie: n
ll.P=Nv
+{ -N6, 1-
seposiTHOCTb (1 -
NfJ),
NO, sepoSITHOCTb NfJ.
(10.16)
Mbt pa3neJIHJIH s11ecb cpenHee uapacTaHHe NfJ H cpnyKryauMu, npHBOJlHlll.He K 38.[laqe 0 CJiy4attuo~ 6Jiym.naHHH
- NB, nepoHTHOCTh ( 1 - NfJ), Av= { 1 -- NO, BeposirttoCTb NB.
(10.16a)
9ro ecTh cnyqati uepaBHhIX rnaros, u MbI npttMeHSieM coorserCTny10w.y10 opMy11y (10.11). Cpe11H11tt KBanpar AJIHHbl wara
4]
179
SPOYHOBO ~BHlKBHMB
paseH
12 = 6.v 2 = (N0) 2 (1 - NB)+ (1 -NO)'l.NfJ =NB (1 - NO),
l 2 =NB. Ha
9TOro COOTHOllleHIUl H ( 10.11) nonyqaeM cpe.nHHtt KB8.ll • par ¢nyKryauHH ua aceM npoMe*yrKe ~ Ha nporHmeHHH Koroporo cpe.a.HHH 9MHCCHH ecrh Nt:
t = nB
(n -
BeJIHKO),
P==Nt=nNO, v 2 = nl 2 == nNB
=
P.
Cpe.nHHn KBa.llpar ¢RyKryauHH a CRyqae nORHOCTblO CRyqanHott 9MHCCMlf B TOl.IHOCTH paBeH cpe.nHeMy qucny 9MHTHpOB3HHb1X qacTHU.
4. Bpoynouo XBHlKenue BepueMc5l K o6cyllnuueHT BR3Koro TpeuHH une .llBHiKeHHH qactHllhl c Maccott m:
w,
[rn. lO
aanuweis ypaaue-
dv+ w·v,
(10.17)
Fx=mdt
Fx osHa4aer cnyqallHy10 ci.u1y, JJ.ettcray10UJ.y10 C"1Jia Fx JJ.OJI)KHa 6htrb cJiyqatlnon, TaK KaK Ha
r JJ.e
Ha qacrwuy. npoTsnKeHHH Jito6oro npoMeiKyTKa npeMeHH (pa3yMHO Bb16paHHOll MaJIOCTH) 6y JleT npoHCXOJlHTb 04eHb 60JiblllOe 4HCJIO cnyqaffHhlX CTOJIKHO· BeHHll 43CTHUbl c MOJieKy JiaJ1H iKHJlKOCTH. PaccMoTpHM Tenepb apeMeHHoii HHTepaan 't, uacTOJibKO MaJlhlfi, q1061>1 H3MeHeI-IHe CKOpOCTH ~'V ll3CTHUbl ua npOTHiKeHHH 3Toro HHTepaaJia 6hIJIO Ma.no. Torna nonHhlf:t HMTIYJihC CHJibl, JiellCTBYIOIUei1 113 11acri-111y 3a epeM51 't, paBeH
sFx dt. "t
X=
(10.18)
0
=0
CKopoCTb qacTHUbl panna v 0 npM t rpwpoeattHe (10.17) no HHTepBaJiy t
m (v 1 -
v 0)
+
M v1 JJ.aer:
WV 0't
npH t
= 't;
HHTe-
= X.
(10.19)
+ X.
(10.20)
fleperpynnHpOabmasi 4J1eH1>1, uMeeM:
mv1 = v 0 (m -W't) TaK KaK
Fx
nonHOCTbIO cnyqattHa, MO/KHO CllHTaTh:
x= HO Ha)lO ytteCTb, 4TO KnanpaT: s( Vi-'Vo 2 _
Ycpenusist
l
X'A
0,
(10.21)
He paaHo
HYnro. BoaaeAeM ( 10.20) B
-m + m2
sareM
2Wt
(10.22)
W
2 2
n
2
't )
LX -r- m2
+ 2X m
YllHTbIBasi,
Vo (
qTo
l
-
Mhl
Wt) (10.22) m '
nono>KHJIH
X
.
.
4)
181
SPOYHOBO 4BH>KBHHB
paBHbJM HyJUO, H8'1)leM cpe)lHee SH a qet-1He
v:: (10.23)
T,
+
Hawa cncTeMa (4acrnua iKHJJ.KOCTh) HMeeT TeMnepa-rypy H K Heii npHMeHHMhl COOTHOllieHHSI KHHel'H4.eCKOfi reopHH:,
FIMeHHo:
l
-
-2 m·v 2 =
I kT. 2
noaTo~y
(10.24) no.nC'f8BJISISI
(10.24)
B
(10.23)
H paspewaH OTHOCH'feJlbHO X~,
nony 11aeM n.-151 a-roti Be.1JHqHHhl
X2 == 2wtk T. YITaK,
D.anee,
cpen.HH'1
KBan.paT
(10.23)
nepenHChlBaSI
l
--;
2 mvi = Mbl
saMetJaeM,
l
lfTO
2
(10.25)
HMny ..1hca
CHJihI
He patteH ttyJiro.
a BHJJ.e
-:rl mv~-- ( 1 ---;n + X~ 2m ' 2W't )
--
mv~
(2wt/tn)
npe.ncTannHeT
norep10
KH ..
HeTHqecKoti 3HeprHH qaCTHUbl npH ee JJ.BHiKeHHH B B513KOti cpeJI.e, 2 TOf JJ.3 K31{ /2m npe.llCTaBJISiel' KOMileUCHpyIDJ..UyIO aueprHIO, JlOCTaBJISieMylO qacTH:lle CJiyltatiHblMH CTOJIKHOBeHHSIMH c Mone ...
X
KrJlaMH. J.1TaK,
Mbl
Ha6J110)laeM
nepeJiaqy
aHeprHH B JJ.Byx HanpaB·
JteHHSIX: KHHernqecKasi atteprHSI qacTHLlhI -+ JJ.BHiKenne ~oJteKy JI. llBHiKeHHe MOJieKyJI-+ KHHeTHqecKaH 3HeprHSI qaCTHUbl. llanee, Mbl BH.ll.HM, qTO eCJIH BSl3KOe TpenHe eeJIHKO, TO H KOMneHcnpy10mnn nepeHOC 3HeprHH B€JIHK. 60JibUIOe BH3Koe TpeHHe 03H3qae'f KOH
CHJlbHYIO
CBH3b Memn.y qaCTHuen H: OKpyiKaIOllJ,efi ee BH3 ..
iKH.llKOCTblO,
a
CJieJlOBaTeJibHO,
H
MOlll.Hble
CTOJIKHOBeHHH,
H:~ny JJbCaM )(. p aoora CJtyqat1HhlX CHJI npoH380JJ.HTC51 sa cqeT 3HeprHif T€IlJIOBOro JJ.BH·
COOTBeTCTBYIOIUHe
60JibWHM
cnyqatiHbIM
'IKeHHSI (T. e. TellJIOThl) H CHOBa npeepa11~aeTCH 8 TellJIOTY qepes nocpeJI.C'fBO BS13KOrO TpeHHH.
182
TBO.TIOBOE 4BUiKHHHE H SPOYHOBO 4BHiKHHHB
IlJI)I
HaXO)l{,lleHIHI
=
0
= 0,
V0
10
cpe,llHeKBa,llpaTHllHOfO CMemeHHSI llaCTMilbI
Mhl HeCKO.TihKO yrrpoCTHM
f
[rn.
f
a npH
=
paccyiK.lleHHe, npeJJ.£10.TIO)l{HB, l!TO rrpH 't V
=
V1. npeJJ.II0.TIO:il01.\f TaKiKe,
llTO
I10CJie MOMeHTa 't: CH.TI bl, ,lletiC'fBYIOUlHe Ha qac-n1uy' OTCYTCTBy10T; MHalle fOBOpH, MbI BhIJJ.eJIHeM OJI.HO H3 CTO.TIKHOBeHH.ti 11 paccMaTp11BaeM
ero,
a
MHJJ.HBHJJ.yaJihHOe
TaKiKe
K8iKJJ.Oe
co6h!THe.
H3
JJ.pyr11x CTOJIKHOBeHHtl KaK
Tor JI.a nocne 't: JJ.BH1KeH11e aaTyxae-r,
M Mbl MO)l{eM sanHcaTb:
wt
m.
Ho TaK
HaXOJJ.HM~
x,
Pa3pelllaH OTHOCHTeJihHO
( 10.26)
KaK 't: Mano, MhI MOIKeM B3HTh rrpeJJ.e.11h1 JJ.JIH :>KcnoHe1-1-
u11aJihHOtl
cpyHKUHl1
0
M
oo;
£109TOMY 113 (
10.20)
rrp11
Vo= 0
nonyqaeM:
_tnVr_X X----.
w
BeJIH411Ha nyJihca
x
X,
npeJJ.craanSieT
nonylleHHoro
0.TIY)KJJ.aHHH,
TaK
x
KaK
a
(10.27)
w
CMeaI,eHYie qacTHUhI BCJieJJ.CTBHe HM-
M0~1eHT 't. 3To ecTh BHJJ. cny 11atlHoro
== 0
I10TOMy, l!TO
B03BOJJ.SI B KBa4paT H ycpe4HHSI (
10.27),
x == 0.
DO,llCTaHOBKa
IlnSI MaJihlX
(10.25)
oonhrnero
B
HHTepB8JIOB
't:,
(10.28)
-2 w.
(10.28)
JJ.aeT:
x2=
2kT tt.
rrpoMe)l{yTKa
X 2 =F O;
nonyqaeM:
- 2_ x2
X
Ho
(10.29)
w
BpeMeHH
t,
coJJ.ep)l{amero MHoro
KBaJJ.paThI CMeUieH11ti cyMM11py10TCSI KaK
He3aBHCHMhJe BeJIHttHHbI, M Mbl HMee.M:
--
x2===
2kTt
w
•
(10.30)
9TOT peaynhTaT OhIJI npoBepeH :>Kcnep11MeHTaJibHO rryTeM Ha6JI10JI.eHMSI
saBHCHMOCTl1
qacrwu B cycneH311H.
OT
BpeMeHM
IlOJIO)l{eHH.t;t
KOJIJI011,l.l,aJibllbIX
5]
183
TBfiJIOBOB ABBmBHHB B 9RBKTPBQBCKott UBUH
5. Tennoeoe nsuHKHO npe.ncTaBl1Tb, qro cneKTp npSI· MoyronhHoro HMny Jibca Bh1C0Tof1 F H JtJIHTeJJbHOCTbIO 6 npoCTHpaerc~ npHOJIH3HTeJlbHO OT_ -VM .no VM c DOCTOHHHOD cne1KHO BSHTb
iKenaeM
= FB.
naoJlHTb
( 11. 7)
0Tp11uaTeJibHble 11acTOThI, ro
(1 LS) H yJJ.BOHTb cneKrpaJibHYfO nJIOTHOCTb (KaK B (11.4)):
C' (v) = 2C (v) = 2F0.
( 11.9)
PaccMOTpHM renepb nocne.QOBaTenbHOCTb HMny.nbcoo AJJHTenb· HOCTbIO 6 u co cpeJl.HHM Fi. CpeJl.HHl-1 KBaJl.paT (11.3) paaeH
E2 =F2 =2RkTin saHHMaer 9TOMy
a.n..c.
cor .nac110
(11.10)
npn6nusHrenbHO nonocy 'iaCTOT 0 ~ 'i ~ vM· Do·
e = 4RkT'l.M-
(ll.1'1)
21
189
METOJI. HADKBHCT A
pacnpeJI.eJieHa noqTH paBHOMepHO B I10JIOCe ttaCTOT OT 0 JI.0 vM· Mo)f(HO cJI.enarb nponopuwoHaJibHhIM nepexoJl K Me1-1hrneMy HH· Tepeany qacTOT Av, T. e. K ffHTepeany Me:ao fJ)
Bsensi 11oopo1'HOCTb KoHrypa ro L
0 Q o=R,
(11.24) (11.25)
ff HHeTH4fCKaSI 9Hepr0H
B KOHType npH HeKoropoft qaCTOTe
v
paatta
(11.26) fI011HaR
KHHeTH11ecKaR
cyMMHpoeaHHR no eceM 33MfHHfTCSl HHTerpaJtOM
·- !-En,._=
Ekin=
9nepr0H
MoJKer
4acroraM
or v
OblTh
=0
nony4eHa nyTeM
.n.o
"= oo;
cyMMa
o::>
~ 1
72" kT Q0 ~ l.cyl1._=---
. . . ....
TC
Wo
dw Q:.i( l + _o
UJ _
wt4>
2)1 •
CJ>o KHbie npHMeHeHHH TepMOJll1HBMHKH K HeoopaT111\lblM npoueccaM 3). 1)
H. A. Lorentz, Les theories statistiques en thermodynamique, Teubner, Leipzig, 1916; G. L. de Haas-Lorentz, Die Brownsche Bewegung, Vieweg, Bra unschweig, l 931. :!) H. B. Ca 11 en, Ph. D. thesis MIT, 1948; R. F. G re en e, Ii. B. Callen, Phys. Rev. 83, 1231 (1951); H.B. Callen, T. A. Welton, ibid. 83, 34 (1951); H.B. Callen, R. F. Greene, ibid. 86, 702 (1952). 8 ) L. 0 n sager, Phys. Rev. 37, 405 (1931); 38, 2265 (1931); H. B. G. Casimir, Rev. Mod. Phys. 17, 343 (1945); C. P. .Lte r poor, TepMO.llHHaMHKa Heo6parHMb1.X. npoueccoB, nepeBO.ll c aHrJIHHCKoro, rocrex11a.aa r, 1956.
41
195
OBOBWEHUSI ~OPMYnbl HAAKBUCTA Ilpu
olleHh BhlCOKHX
qacroTax
c1,rryau11S1
HJleTCSL flpe>KJle ecero, uyncuo yqeCTb, neHH caooo,ll,hl
Moryr ttallaTb
11rparb
cymecreeuuo Me ..
4TO MHOrHe HOBhie CTeaaMeTHYIO
po11b.
Tai1npHMHTeJib BKJII04eH nocneJiosareJihHO c couporHB.11eHHeM
R
AKr0e11b1e
11 peaKTHBHOCThIO conpoT11eneH11H
X,
KaK
noKa3aHo
1 1.5.
Ha pwc.
B uen0 ( conpo-r0eneH11e
R,
a -ra10Ke
PHc. 11..'i AKtMBHoe conporus:1eHHe R, pcaKTHBHOCTb )( 11 BblnpHMHteJib AB B nocJie.lloBa reJibHoM coe)lHHeHHH. l.UyM renepHpyercH B conpotHBJICHHH. l1McetCH BbHipHMJICHHaH 9.JJ..C., HO Het BbIOpHMJieHHOfO TOK3. aKTHBHOe conpOTHBJieHHe
BhlnpHMHTeJIH) 6y.u.yT C03JiaBaTb CJJY·
qaf:iHhle 3.JI.C., 06ycnOBJieHH&Ie TennoehlM JIBH)KeH0eM. Bhl11pHM11TeJJh He MO)f(eT BblnpHMHTb cnyqaf:iHble :J •
II
.u..c.,
TOKH, Bbl3BaHHbie 9THMl1
Tai( KaK BhlllpHMJieHHblti TOK
Mor 6h1 npoHsBeCTH pa6oTy eonpeKH BTOpOMY Hall.any TepMOJJ.HHaMHKH. XapaKTep0ct0Ka noKasatta OHa TO
l
0.u.eT MO)f(HO
Ha
p0c.
BhlnpHMHreJIH
11.6;
TaK
KaK
Kpyqe B npaBOti 4acTH, 3aKJIIOll.Hlh, ll.TO
cf>nyI
